User:Geometry guy/Calculus

From Wikipedia, the free encyclopedia

This page is about derivatives and differentiation in mathematical calculus. For other uses, see Geometry guy/Calculus (disambiguation).

For a non-technical overview of the subject, see Calculus.

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.

In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. The derivative of a function at a chosen input value describes the behavior of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In general, the derivative of a function at a point determines the best linear approximation to the function at that point.^[1]

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.

Differentiation has applications to all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. By applying game theory, differentiation can even provide best strategies for competing corporations.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear throughout mathematics, in fields such as complex analysis, functional analysis, differential geometry, and even abstract algebra.

[edit] Differentiation and the derivative

Differentiation is a method of computing how the change in a quantity x determines the change in a dependent quantity y. The result of this computation is called the derivative of y with respect to x. The derivative gives a linear approximation to the actual dependence of y on x, meaning that this approximation assumes that a change in x will produce a proportional change in y. The derivative is, in a precise sense, the best possible linear approximation to the actual dependence of y on x. Each choice of a specific value for x can have a different derivative, because the way in which y changes can be different for different x.

More precisely, the dependency of y on x means that y is a function of x. This relationship is often denoted y = f(x), where f denotes that function. In the one-dimensional case, when x and y are real numbers, the derivative is a real number. In higher dimensions, the derivative is a linear transformation. The derivative can also be extended to more exotic situations, such as infinite dimensions and fractals.

[edit] The derivative in the single variable case

Main article: Derivative (single variable)

The tangent line at (x, f(x))

Suppose that x and y are real numbers, so that the graph of function f can be drawn in the plane. Choose a value of x, for example x = 2. Because f is a function, that choice of x-value determines a y-value f(x), for example f(2) = 1. Lines in the plane correspond to possible linear approximations of f. For a line to be a good linear approximation near the x-value 2, it should pass through the point (2, 1). However, not all of the lines passing through (2, 1) are good approximations. The derivative of f at 2 is equal to the slope of the line through (2, 1) which follows f most closely. This line is called the tangent line, so the derivative of f at 2, written f'(2), is equal to the slope of the tangent line to the graph of f at (2, 1). The present article will assume that the tangent line exists; for information on functions which do not have tangent lines, see the main article on single-variable derivatives.

The secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h)).

While it is easy to see the tangent line on a graph, it is difficult to compute the derivative precisely with this method. However, it is easy to compute an approximation for the derivative. This is done by using secant lines. A secant line is a line which passes through two (or more) points of the graph of f. If the two points are close together, for instance, $(2,1)$ and $(2.1,1.1)$ , then the secant line between these points is close to the tangent line at $(2,1)$ . In particular, their slopes will be very similar, so the slope of the secant line gives an approximation to the slope of the tangent line. Because the slope of the secant line is the difference in the y-values of the two points divided by the difference in the x values of the two points, it is easy to compute:

$\frac{f(2.1) - f(2)}{2.1 - 2} = \frac{1.1 - 1}{2.1 - 2} = \frac{0.1}{0.1} = 1.$

The tangent line as limit of secants.

The approximation given by the secant line between $(2,1)$ and $(2.1,1.1)$ is probably not equal to the slope of the tangent line. The secant line between $(2,1)$ and a closer point, such as $(2.01,1.005)$ may be closer to the tangent line, so it may give a better approximation. An even closer point, such as $(2.001,1.0003)$ may give a still better approximation. In general, the nearer the second point is to $(2,1)$ , the better the approximation. To make this precise, write the x-value of the nearby point as $2 + h$ , where h is a number very close to zero. The slope of secant line between $(2,1)$ and $(2 + h, f (2 + h))$ is approximately equal to the slope of the tangent line:

$f'(2) \approx \frac{f(2 + h) - f(2)}{(2 + h) - 2} = \frac{f(2 + h) - 1}{h}.$

These fractions are called difference quotients. When h is very close to zero, these difference quotients are very close to the slope of the tangent line. The nearer that h is to zero, the nearer the difference quotient is to the slope of the tangent line. The slope of the tangent line is characterized by this property, meaning that it is the unique number which the difference quotients approximate. In other words, finding all the possible slopes of secant lines determines what the slope of the tangent line must be. The process of going backwards from knowing approximations to finding the approximated number is called taking a limit. In symbols,

$f'(2) = \lim_{h \rightarrow 0} \frac{f(2 + h) - f(2)}{(2 + h) - 2} = \lim_{h \rightarrow 0} \frac{f(2 + h) - 1}{h}.$

To see that the derivative is actually the best linear approximation, rewrite this expression as

$\lim_{h \rightarrow 0} \frac{f(2 + h) - f(2) - f'(2)h}{h} = 0.$

This implies that

$\lim_{h \rightarrow 0} f(2 + h) - f(2) - f'(2)h = 0,$

so that

$f(2 + h) \approx f(2) + f'(2)h.$

After making the substitution x = 2 + h, this becomes

$y = f(x) \approx 1 + f'(2)(x - 2).$

After subtracting 1 from both sides, this becomes the point-slope form of the line through $(2,1)$ with slope f'(2). Consequently, this line is the tangent line.

[edit] The single-variable derivative as a linear transformation

The linear approximation defined by the derivative assumes that a change in the value of x produces a proportional change in the value of y. For example, suppose that in the example above, f′(x) equals 1/4. Then the equation for the tangent line is

$y = 1 + \frac{1}{4}(x - 2).$

When x is 2, y is 1. When x is changed by 1 to be 3, then y is changed by 1/4 to be 5/4. When x is changed by 2, which is twice one, then y is changed by 1/2, which is twice 1/4, to be 3/2. When x is changed by 3, which is thrice one, then y is changed by 3/4, which is thrice 1/4, to be 7/4.

Consequently the information contained in the tangent line approximation can be expressed in terms of vectors originating at x = 2. Vectors are a precise way of describing how to travel from one point to another, because they specify a length and a direction. A vector of length one pointing in the positive direction corresponds to the change from x = 1 to x = 2. It determines a vector of length 1/4 corresponding to the change from y = 1 to y = 5/4. Similarly, a vector of length 2.5 corresponds to the change from x = 1 to x = 3.5 and determines a vector of length 5/8 corresponding to the change from y = 1 to y = 13/8. The zero vector corresponds to no change in x, and it determines a zero-length vector which corresponds to no change in y.

These vectors are called tangent vectors. The tangent vector of length one in the positive x-direction will be written $\vec{x}$ . (In more advanced works it is frequently written d/dx or ∂/∂x, but to avoid confusion this article will avoid this notation.) Similarly, the tangent vector of length one in the positive y-direction will be written $\vec{y}$ . The tangent line approximation turns vectors in the x-direction into vectors in the y-direction. It turns $\vec{x}$ into $(1/4)\vec{y}$ , $2\vec{x}$ into $(1/2)\vec{y}$ , $3.5\vec{x}$ into $(5/8)\vec{x}$ , and so on. This transformation is called the pushforward at x = 2 and it is written $f * (2)$ . Even though $f * (2)$ looks like a function that has already been evaluated, it is still a function which takes vectors in the x-direction and returns vectors in the y-direction. In symbols, therefore,

$f_*(2)(\vec{x}) = (1/4)\vec{y},$

$f_*(2)(2\vec{x}) = (1/2)\vec{y},$

$f_*(2)(3.5\vec{x}) = (5/8)\vec{y}.$

The first line would be read as, "The pushforward of $\vec{x}$ is $(1/4)\vec{y}$ ." Rewriting these in terms of the derivative of f gives the formulas

$f_*(2)(\vec{x}) = f'(2)\vec{y},$

$f_*(2)(2\vec{x}) = (f'(2) \cdot 2)\vec{y},$

$f_*(2)(3.5\vec{x}) = (f'(2) \cdot 3.5)\vec{y}.$

These equations are true because $f *$ is a linear transformation. This means that it transforms the sum of two vectors into the sum of their transforms, and it transforms a scaled vector into the scale of the transformed vector. For instance, it implies the following equalities:

$f_*(2)(\vec{x} + \vec{x}) = f_*(2)(\vec{x}) + f_*(2)(\vec{x}),$

$f_*(2)(3.5\vec{x}) = 3.5f_*(2)(\vec{x}).$

In the one-variable case, preservation of scaling implies preservation of sums. This is not true when f is a function of several variables.

$f * (2)$ and f'(2) provide the same information when f is a function of one variable. This is because the inputs to f can change in only one direction, namely the x-direction, so the amount by which $f * (2)$ scales $\vec{x}$ is determined by a single number, f'(2). When f is a function of several variables, then a single number no longer suffices to describe the best linear approximation of f.

[edit] Derivatives of vector valued functions

A vector-valued function γ(t) of a real variable is a function which sends real numbers to vectors in some vector space Rⁿ. These define parametric curves. For example, if t denotes time, then γ(t) might be the x and y-coordinates of a moving particle. γ(t) might be the function

$\gamma(t) = \left(\frac{1-t^2}{1 + t^2}, \frac{2t}{1 + t^2}\right),$

which defines a circle in the plane. The derivative of γ is the best linear approximation to γ, meaning that it defines the line whose behavior most closely follows the behavior of γ at some chosen t. As above, this line is called the tangent line to γ. The tangent line is also a parametric curve. To determine the tangent line to γ at a point, for example at t = 2, split γ into its coordinate functions:

$\gamma_1(t) = \frac{1-t^2}{1+t^2},$

$\gamma_2(t) = \frac{2t}{1 + t^2},$

γ(t) = (γ 1 (t),γ 2 (t)).

Because the tangent line follows γ as closely as possible in the x and y-directions, it must be determined by the tangent lines to $γ 1$ and $γ 2$ . In other words, the best linear approximation to γ in the x-direction is determined by the best linear approximation to the x-part of γ, that is, by $γ 1$ . The same is true for the y-direction. Because $γ 1$ and $γ 2$ are real-valued functions of one real variable, their tangent lines at t = 2 have slopes equal to their derivatives at t = 2. These two derivatives determine a vector called the derivative of γ at t = 2:

γ'(2) = (γ 1'(2),γ 2'(2)).

Each of these coordinates on its own is the slope of a tangent line. These slopes are directions. $γ 1'(2)$ , for example, specifies that a one unit change in t changes the x coordinate of the tangent line to $γ 1$ at t = 2 by $γ 1'(2)$ . Together, $γ 1'(2)$ and $γ 2'(2)$ specify that for a change of one unit in t, the tangent line to γ at t = 2 changes its x-coordinate by $γ 1'(2)$ and its y-coordinate by $γ 2'(2)$ . Consequently, the tangent line can be written as the parametric curve

T (t) = (γ 1'(2)(t - 2),γ 2'(2)(t - 2)).

In general, if γ(t) is a vector-valued function with values in Rⁿ, then its derivative at the point t₀ is the vector

$\gamma'(t_0)=\lim_{h\to 0}\frac{\gamma(t_0+h) - \gamma(t_0)}{h},$

if the limit exists. The subtraction in the numerator is subtraction of vectors, not scalars.

The derivative of γ defines a linear transformation of tangent vectors. It takes tangent vectors in the t-direction, such as the unit tangent vector $\vec{t}$ , to tangent vectors in the target space Rⁿ. Because all tangent vectors in the t-direction can be found by rescaling $\vec{t}$ , all of the tangent vectors in the image of $γ * (2)$ can be found by rescaling the tangent vector $\gamma_*(2)(\vec{t})$ . By definition this vector is $(γ 1'(2),γ 2'(2))$ , the derivative of γ at t = 2. For this reason, the derivative of γ is also called "the tangent vector" to γ.

[edit] Partial derivatives

Main article: Partial derivative

Suppose that f is a real-valued function that depends on more than one variable. For instance,

z = f (x, y) = x 2 + x y + y 2 .

Choose a direction, say the y-direction. The partial derivative of f in the y-direction is the best linear approximation to f in the y-direction. Geometrically, it corresponds to a line which travels in the y-direction (meaning that it has fixed x-coordinate) and which follows f as closely as possible in the y-direction. This does not take into account the behavior of f in the x-direction, so, like the ordinary derivative of a single-variable function, it is represented by a single number.

For instance, here is what the partial derivative of f in the y-direction at the point $(5, - 2)$ means. First, reinterpret f as a family of functions of one variable indexed by the other variables:

f (x, y) = f x (y) = x 2 + x y + y 2 .

In other words, every value of x chooses a function, denoted f_x, which is a function of one real number. That is,

$x \mapsto f_x,$

f x (y) = x 2 + x y + y 2 .

To compute the partial derivative at $(5, - 2)$ , choose x to be 5. Then f(x,y) determines a function f₅ which sends y to 5² + 5y + y²:

f 5 (y) = 25 + 5 y + y 2 .

Because 5 is a constant, not a variable, f₅ is a function of only one real variable. Consequently the definition of the derivative for a function of one variable applies, and the partial derivative of f in the y-direction at the point $(5, - 2)$ is defined to be $f 5'( - 2)$ . This is a single-variable derivative, and it has an interpretation as a limit of slopes of secant lines. In the present example, these secant lines are lines between the two points $(5, - 2,19)$ and $(5, - 2 + h,19 + 2 h + h 2)$ , where h is a small real number. Notice that the x-coordinates of these two points are the same, so that these secant lines only measure behavior in the y-direction. Applying the definition of the single variable derivative gives an expression for the partial derivative:

$\frac{\part f}{\part y}(5, -2) = f_5'(-2) = \lim_{h \rightarrow 0} \frac{f_5(-2+h) - f_5(-2)}{h} = \lim_{h \rightarrow 0} \frac{f(5, -2+h) - f(5, -2)}{h}.$

Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee". The notation (∂f/∂y)(5, -2) stands for the partial derivative of f in the y-direction at the point $(5, - 2)$ . By the rules below for computing the derivative, $f 5'( - 2)$ equals 1.

Similarly to derivatives of single-variable functions, partial derivatives determine linear transformations of tangent vectors. In the example above, the partial derivative in the y-direction determines a linear transformation from vectors in the y-direction to vectors in the z-direction. These tangent vectors are multiples of the unit vector $\vec{y}$ . The partial derivative in the y-direction does not determine how to transform other tangent vectors, such as $\vec{x}$ . Instead, the partial derivative in the x-direction determines how to transform $\vec{x}$ . However, no single partial derivative determines how to transform $\vec{x} + 3\vec{y}$ , because it does not point in the same direction as either of the coordinate axes.

[edit] Directional derivatives

Main article: Directional derivative

Let f be the same function as in the previous section, that is,

z = f (x, y) = x 2 + x y + y 2 .

The partial derivatives of f determine the best linear approximation of f in the x-direction and the y-direction. They do not, however, directly find the best linear approximation in any other direction, such as along a diagonal. These are found using directional derivatives. To find the best linear approximation to f in some other direction, choose a vector pointing in that direction, say for example $\vec{v} = (1, 3)$ . The directional derivative in the direction $\vec{v}$ is the best linear approximation to f along the line determined by the vector v. Like single-variable derivatives and partial derivatives, it is the limit of slopes of secant lines. For instance, consider again the point $(5, - 2)$ . To find the directional derivative at $(5, - 2)$ in the direction $\vec{v}$ , take secant lines from the point $(5, - 2,19)$ to another point which lies along the direction given by $\vec{v}$ . Since $\vec{v} = (1, 3)$ , these points have the form

(5 + h, - 2 + 3 h,(5 + h) 2 + (5 + h)( - 2 + 3 h) + ( - 2 + 3 h) 2).

Using the usual formula for slope, therefore, the directional derivative is

$D_{(1,3)}f(5, -2) = \lim_{h \rightarrow 0} \frac{f(5 + h, -2 + 3h) - f(5, -2)}{h}.$

If $\vec{v}$ is not $(1,3)$ but is instead a unit vector in the direction of one of the coordinate axes, that is, $(1,0)$ or $(0,1)$ , then the directional derivative in the $\vec{v}$ -direction is a partial derivative.

Let λ be a scalar. The substitution of h/λ for h changes the $\lambda\vec{v}$ direction's difference quotient into λ times the $\vec{v}$ direction's difference quotient. Consequently, the directional derivative in the $\lambda\vec{v}$ direction is λ times the directional derivative in the $\vec{v}$ direction. Because of this, directional derivatives are often considered only for unit vectors $\vec{v}$ .

The directional derivative in the $\vec{v}$ -direction determines a linear transformation of tangent vectors in the $\vec{v}$ -direction. This is entirely analogous to how the partial derivative in the y-direction determines a linear transformation of tangent vectors in the $\vec{y}$ -direction. In the example above, for instance, the directional derivative determines how to transform the tangent vector $\vec{x} + 3\vec{y}$ .

[edit] The total derivative and the Jacobian

Main article: Total derivative

Suppose that f is a function that depends on several variables. For instance, suppose that f is

(u, v, w) = f (x, y) = (x 2 + x y + y 2, x 3 - y 3, x y).

The total derivative of f at some point, for example $(5, - 2)$ again, is the best linear approximation to f. Unlike partial derivatives, it is the best linear approximation in all directions simultaneously. Also unlike partial derivatives, it is not a number. This is because it must be the best linear approximation in two directions at once, and a single number does not contain enough information to approximate a function with two or more inputs. Instead, the total derivative is a higher-dimensional analog of the pushforward $f *$ . To be precise, the total derivative of f at $(5, - 2)$ is defined to be the unique linear transformation f'(5, -2) from R² to R³ such that

$\lim_{||\mathbf{h}||\rightarrow 0} \frac{||f((5, -2) +\mathbf{h}) - f(5, -2) - f'(5, -2)\mathbf{h}||}{||\mathbf{h}||} = 0.$

Here h is a vector in R², so the norm in the denominator is the standard length on R². However, f'(5, -2)h is a vector in R³, and the norm in the numerator is the standard norm on R³. Notice also that while the notation for the total derivative is f'(5, -2), it is more analogous to the pushforward, which would be written f_*(5, -2).

If the total derivative exists at (5, -2), then all the partial derivatives of f exist at (5, -2). Geometrically, this means that if f has a best linear approximation, then f has a best linear approximation in each direction. Rewrite f using coordinate functions, so that f = (f₁, f₂, f₃). Then the total derivative can be expressed as a matrix called the Jacobian matrix of f at (5, -2):

$f'(5, -2) = \text{Jac}_{(5, -2)} = \begin{pmatrix} \frac{\partial f_1}{\partial x}(5, -2) & \frac{\partial f_1}{\partial y}(5, -2) & \frac{\partial f_1}{\partial z}(5, -2) \\ \frac{\partial f_2}{\partial x}(5, -2) & \frac{\partial f_2}{\partial y}(5, -2) & \frac{\partial f_2}{\partial z}(5, -2) \\ \frac{\partial f_3}{\partial x}(5, -2) & \frac{\partial f_3}{\partial y}(5, -2) & \frac{\partial f_3}{\partial z}(5, -2) \\ \end{pmatrix}.$

The existence of all the partial derivatives does not imply the existence of the total derivative. This means that there are functions which have a best linear approximation in every direction but no best linear approximation in all directions simultaneously. However, if all the partial derivatives exist and satisfy a mild smoothness condition, then the total derivative exists.

Because the Jacobian matrix is made up of the partial derivatives of f, knowing the Jacobian matrix determines all the partial derivatives of f. In fact, the Jacobian matrix even determines all the directional derivatives of f. The directional derivative of f in the direction v is equal to

$D_{\mathbf{v}}{f}(\boldsymbol{x}) = \sum_{j=1}^n v_j \frac{\partial f}{\partial x_j}.$

This is because derivatives are linear transformations; the formula says that the directional derivative in the direction v is determined by splitting v into its component directions, finding the partial derivatives in those directions, and then scaling those directions by the lengths of v in those directions.

The tangent line to the graph of a function generalizes to the tangent space. The tangent space is a geometric representation of the best linear approximation to the graph of a function.

The transpose of the Jacobian matrix determines a linear map from R^m to Rⁿ. More intrinsically, this is the dual map on dual vector spaces. This linear map is called the pullback. In the example above, the pullback would be written f^*(5, -2).

The definition of the total derivative subsumes the definition of the derivative in one variable. In this case, the total derivative exists if and only if the usual derivative exists. The Jacobian matrix at the point a reduces to a 1×1 matrix whose only entry is the derivative f'(a). This 1×1 matrix satisfies the property that f(a + h) - f(a) - f'(a)h is approximately zero, in other words that

$f(a+h) \approx f(a) + f'(a)h.$

Up to changing variables, this is the statement that the function $x \mapsto f(a) + f'(a)(x-a)$ is the best linear approximation to f at a.

[edit] The derivative as a function

Let f be a real-valued function of a real number, such as f(x) = x². For every choice of x, f has a derivative. For instance, when x = 0, the derivative is zero; when x = 1, the derivative is 2; and when x = 3, the derivative is 6. These derivatives can be assembled together into a function, called the derivative function or just the derivative of f. This function is written f'(x). Its domain is all real numbers, and at each point in its domain, the value of the function is the derivative of f at that point. In other words, the derivative of f collects all the derivatives of f at all the points in the domain of f. For the function f(x) = x², the derivative f'(x) equals the doubling function g(x) = 2x.

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions which have derivatives at every point of their domain and whose range is a set of functions. Call this operator D. Then D(f) is the function f'(x). Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f'(a).

For comparison, consider the squaring function f(x) =x². f is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs:

$1 \mapsto 1,$

$2 \mapsto 4,$

$3 \mapsto 9.$

The operator D, however, is not defined on individual numbers. It is only defined on functions:

$(x \mapsto 1) \mapsto (x \mapsto 0),$

$(x \mapsto x) \mapsto (x \mapsto 1),$

$(x \mapsto x^2) \mapsto (x \mapsto 2x).$

Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to the squaring function f(x), D outputs the doubling function g(x). The doubling function can then be evaluated as usual: g(1) = 2, g(2) = 4, and so on.

This idea extends to derivatives of vector-valued functions, partial derivatives, and directional derivatives. In all of these cases, the derivative function takes a point in the domain of the original function and returns the derivative of the function at that point. This is a number or a vector, depending on whether the original function returned a number or a vector.

However, the total derivative of a function does not give another function in the same way that one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target. The tangent bundle is a space built from the original space and all of its tangent vectors. A point in the tangent bundle is an ordered pair (x, v), where x is a point in the original space and v is a tangent vector at that point. The original function determines how to send x to a point of the target space, and the total derivative determines how to transform v.

[edit] Higher derivatives

Let f be a differentiable function, and let f'(x) be its derivative. f'(x) is a function, so it might have a derivative. If it does, then the derivative of f'(x) is written f''(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if it exists, is written f'''(x) and is called the third derivative of f. These repeated derivatives are called higher-order derivatives.

A function f need not have a derivative, for example, if it is not continuous. Similarly, even if f does have a derivative, it may not have a second derivative. For example, let

$f(x) = \begin{cases} x^2, & \mbox{if }x\ge 0 \\ -x^2, & \mbox{if }x \le 0\end{cases}$ .

An elementary calculation shows that f is a differentiable function whose derivative is

$f'(x) = \begin{cases} 2x, & \mbox{if }x\ge 0 \\ -2x, & \mbox{if }x \le 0\end{cases}$ .

f'(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that a function can have k derivatives for any non-negative integer k but no (k + 1)-order derivative. A function that has k successive derivatives is called k times differentiable. If in addition the kth derivative is continuous, then the function is said to be of differentiability class C^k. (This is a stronger condition than having k derivatives. For an example, see differentiability class.) A function that has infinitely many derivatives is called infinitely differentiable or smooth.

On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.

The derivatives of a function f at a point x provide polynomial approximations to that function near x. For example, if f is twice differentiable, then

$f(x+h) \approx f(x) + f'(x)h + \tfrac12 f''(x) h^2$

in the sense that

$\lim_{h\to 0}\frac{f(x+h) - f(x) - f'(x)h - \frac12 f''(x) h^2}{h^2}=0.$

If f is infinitely differentiable, then this is the beginning of the Taylor series for f.

[edit] Notations for differentiation

Main article: Notation for differentiation

[edit] Leibniz's notation

Main article: Leibniz's notation

The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still commonly used when the equation y=f(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by

$\frac{dy}{dx},\quad\frac{d \bigl(f(x)\bigr)}{dx},\;\;\mathrm{or}\;\; \frac{d}{dx}\bigl(f(x)\bigr).$

Higher derivatives are expressed using the notation

$\frac{d^ny}{dx^n},\quad\frac{d^n\bigl(f(x)\bigr)}{dx^n},\;\;\mathrm{or}\;\;\frac{d^n}{dx^n}\bigl(f(x)\bigr)$

for the nth derivative of y=f(x) (with respect to x).

With Leibniz's notation, we can write the derivative of y at the point x=a in two different ways:

$\frac{dy}{dx}\left.{\!\!\frac{}{}}\right|_{x=a} = \frac{dy}{dx}(a).$

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember^[2]:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$

[edit] Lagrange's notation

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange and uses the prime mark, so that the derivative of a function $f(x)\,$ is denoted $f'(x)\,$ or simply $f'\,$ . Similarly, the second and third derivatives are denoted $(f')'=f''\,$ and $((f')')'=f'''\,$ . Beyond this point, some authors use Roman numerals such as $f^{IV}\,$ for the fourth derivative, whereas other authors place the number of derivatives in parentheses: $f^{(4)}\,$ in this case. The latter notation generalizes to yield the notation $f^{(n)}\,$ for the nth derivative of f.

[edit] Newton's notation

Main article: Newton's notation

Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a derivative. If y = f(t), then $\dot{y}$ and denotes the first derivative of y with respect to t and $\ddot{y}$ denotes the second derivative. This notation is used almost exclusively for time derivatives, meaning that the independent variable of the function represents time. It is very common in physics and in mathematical disciplines connected with physics such as differential equations. While the notation becomes unmanageable for high-order derivatives, in practice only very few derivatives are needed.

[edit] Euler's notation

Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. The second derivative is denoted D²f, and the nth derivative is denoted Dⁿf.

If y=f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written $D x y$ or $D x f (x)$ , although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression.

Euler's notation is useful for stating and solving linear differential equations.

[edit] Computing the derivative

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. For some examples, see Derivative (examples). In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.

[edit] Rules for finding the derivative

Main article: Differentiation rules

In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules. Some of the most basic rules are the following.

Constant rule: if f(x) is constant, then

$f' = 0 \,$

Linearity:

$(af + bg)' = af' + bg' \,$ for all functions f and g and all real numbers a and b.

Product rule:

$(fg)' = f 'g + fg' \,$ for all functions f and g.

Chain rule: If $f (x) = h (g (x))$ , then

$f'(x) = h'(g(x)) g'(x) \,$ .

[edit] Derivatives of elementary functions

Main article: Table of derivatives

In addition, the derivatives of some common functions are useful to know.

Derivatives of powers: if $f (x) = x r$ , where r is any real number, then

f'(x) = r x r - 1,

wherever this function is defined.

For example, if r = 1/2, then f'(x) = (1/2)x^−1/2 is defined only for non-negative x. When r = 0, this rule recovers the constant rule.

Exponential and logarithm functions:

$\frac{d}{dx}\exp(x) = \exp(x).$

$\frac{d}{dx}\ln(x) = 1/x.$

Trigonometric functions:

$\frac{d}{dx}\sin(x) = \cos(x).$

$\frac{d}{dx}\cos(x)= -\sin(x).$

[edit] Example computation

The derivative of

$f(x) = x^4 + \sin (x^2) - \ln(x) e^x + 7\,$

$\begin{align} f'(x) &= 4 x^{(4-1)}+ \frac{d\left(x^2\right)}{dx}\cos (x^2) - \frac{d\left(\ln {x}\right)}{dx} e^x - \ln{x} \frac{d\left(e^x\right)}{dx} + 0 \\ &= 4x^3 + 2x\cos (x^2) - \frac{1}{x} e^x - \ln(x) e^x. \end{align}$

Here the second term was computed using the chain rule and third using the product rule: the known derivatives of the elementary functions x², x⁴, sin(x), ln(x) and exp(x) = e^x were also used.

[edit] History of differentiation

Main article: History of calculus

The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BCE), Archimedes (c. 287 BCE – 212 BCE) and Apollonius of Perga (c. 262 BCE – c. 190 BCE)^[3]. Archimedes also introduced the use of infinitesimals, although these were primarily used to study areas and volumes rather than derivatives and tangents — see Archimedes' use of infinitesimals.

The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 CE, when the astronomer and mathematician Aryabhata (476 – 550) used infinitesimals to study the motion of the moon^[4]. The use of infinitesimals to compute rates of change was developed significantly by Bhaskara (1114-1185): indeed, it has been argued^[5] that many of the key notions of differential calculus can be found in his work.

The modern development of calculus is usually credited to Isaac Newton (1643 – 1727) and Gottfried Leibniz (1646 – 1716), who provided independent^[6] and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes, which had not been significantly extended since the time of Archimedes^[7]. For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Isaac Barrow (1630 – 1677), René Descartes (1596 – 1650), Christiaan Huygens (1629 – 1695), Blaise Pascal (1623 – 1662) and John Wallis (1616 – 1703). In particular, Isaac Barrow is often credited with the early development of the derivative^[8]. Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today.

Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789 – 1857), Bernhard Riemann (1826 – 1866), and Karl Weierstrass (1815 – 1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.

[edit] Applications of derivatives

[edit] Maxima, minima and critical points

If f is a differentiable function on R (or an open interval) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero; points where f '(x) = 0 are called critical points or stationary points (and the value of f at x is called a critical value). (The definition of a critical point is sometimes extended to include points where the derivative does not exist.) Conversely, a critical point x of f can be analysed by considering the second derivative of f at x:

if it is positive, x is a local minimum;
if it is negative, x is a local maximum;
if it is zero, then x could be a local minimum, a local maximum, or neither. (For example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there.)

This is called the second derivative test. An alternative approach, called the first derivative test, involves considering the sign of the f ' on each side of the critical point.

Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in optimization. By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.

This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points.

In higher dimensions, a critical point of a scalar valued function is a point at which the gradient is zero. The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is inconclusive.

[edit] Physics

Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and evolve over time, and the concept of the "time derivative" — the rate of change over time — is essential for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

velocity is the derivative (with respect to time) of an object's displacement (distance from the original position)
acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position.

For example, if an object's position on a line is given by

$x(t) = -16t^2 + 16t + 32 , \,\!$

then the object's velocity is

$\dot x(t) = x'(t) = -32t + 16, \,\!$

and the object's acceleration is

$\ddot x(t) = x''(t) = -32, \,\!$

which is constant.

[edit] Generalizations

Main article: Derivative (generalizations)

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point.

An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. However, this innocent definition hides some very deep properties. If C is identified with R² by writing a complex number z as x + i y, then a differentiable function from C to C is certainly differentiable as a function from R² to R² (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the Cauchy Riemann equations — see holomorphic functions.

Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space which can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R³. The derivative (or differential) of a (differentiable) map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses — see pushforward (differential) and pullback (differential geometry).

Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gâteaux derivative, and of the differential, called the Fréchet derivative.

One deficiency of the classical derivative is that not very many functions are differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all continuous functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average".

The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra.

[edit] Notes

^ Differential calculus, as discussed in this article, is a very well-established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.
^ In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors. Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx. Others define "dx" as an independent variable, and define du by du = dx•f '(x). In non-standard analysis du is defined as an infinitesimal. It is also interpreted as the exterior derivative du of a function u. See differential (infinitesimal) for further information.
^ See Euclid's Elements, The Archimedes Palimpsest and O'Connor, John J. & Robertson, Edmund F., “Apollonius of Perga”, MacTutor History of Mathematics archive
^ O'Connor, John J. & Robertson, Edmund F., “Aryabhata the Elder”, MacTutor History of Mathematics archive
^ Ian G. Pearce. Bhaskaracharya II.
^ Newton began his work in 1666 and Leibniz began his in 1676. However, Leibniz published his first paper in 1684, predating Newton's publication in 1693. It is possible that Leibniz saw drafts of Newton's work in 1673 or 1676, or that Newton made use of Leibniz's work to refine his own. Both Newton and Leibniz claimed that the other plagiarized their respective works. This resulted in a bitter controversy between the two men over who first invented calculus which shook the mathematical community in the early 18th century.
^ This was a monumental achievement, even though a restricted version had been proven previously by James Gregory (1638 – 1675), and some key examples can be found in the work of Pierre de Fermat (1601 – 1665).
^ Eves, H. (1990).

[edit] References

[edit] Print

Anton, Howard; Bivens, Irl & Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single and Multivariable (8th ed.), New York: Wiley, ISBN 978-0471472445
Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0471000051
Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, vol. 1 (2nd ed.), Wiley, ISBN 978-0471000075
Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN 978-0030295584
Larson, Ron; Hostetler, Robert P. & Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, ISBN 978-0618606245
Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0914098898
Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0534393397
Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0312185480

[edit] Online books

Crowell, Benjamin (2003), Calculus, <http://www.lightandmatter.com/calc/>
Garrett, Paul (2004), Notes on First-Year Calculus, <http://www.math.umn.edu/~garrett/calculus/>
Hussain, Faraz (2006), Understanding Calculus, <http://www.understandingcalculus.com/>
Keisler, H. Jerome (2000), Elementary Calculus: An Approach Using Infinitesimals, <http://www.math.wisc.edu/~keisler/calc.html>
Mauch, Sean (2004), Unabridged Version of Sean's Applied Math Book, <http://www.its.caltech.edu/~sean/book/unabridged.html>
Sloughter, Dan (2000), Difference Equations to Differential Equations, <http://math.furman.edu/~dcs/book/>
Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus, <http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm>
Wikibooks, Calculus, <http://en.wikibooks.org/wiki/Calculus>

[edit] See also

[edit] External links

WIMS Function Calculator makes online calculation of derivatives; this software also enables interactive exercises.

ADIFF online symbolic derivatives calculator.

Proving Derivatives from First Principles.