Differentiation rules

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This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

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[edit] Elementary rules of differentiation

Unless otherwise stated, all functions will be functions from R to R, although more generally, the formulae below make sense wherever they are well defined.

[edit] Differentiation is linear

Main article: Linearity of differentiation

For any functions f and g and any real numbers a and b.

(af + bg)' = af' + bg'.\,

In other words, the derivative of the function h(x) = a f(x) + b g(x) with respect to x is

h'(x) = a f '(x) + b g '(x).

In Leibniz's notation this is written

\frac{d(af+bg)}{dx} = a\frac{df}{dx} +b\frac{dg}{dx}.

Special cases include:

  • The constant multiple rule
(af)' = a\,f' \,
  • The addition rule
(f + g)' = f' + g'\,
  • The subtraction rule
(f - g)' = f' - g'.\,

[edit] The product or Liebniz rule

Main article: Product rule

For any functions f and g,

(fg)' = f' g + f g'.\,

In other words, the derivative of the function h(x) = f(x) g(x) with respect to x is

h'(x) = f '(x) g(x) + f(x) g '(x).

In Leibniz's notation this is written

\frac{d(fg)}{dx} = \frac{df}{dx} g + f \frac{dg}{dx}.

[edit] The chain rule

Main article: Chain rule

This is a rule for computing the derivative of a function of a function, i.e., of the composite f\circ g of two functions f and g:

(f \circ g)' = (f' \circ g)g'.\,

In other words, the derivative of the function h(x) = f(g(x)) with respect to x is

h'(x) = f '(g(x)) g '(x).

In Leibniz's notation this is written (suggestively) as:

\frac{df}{dx} = \frac{df}{dg} \frac{dg}{dx}.\,

[edit] The polynomial or elementary power rule

Main article: Calculus with polynomials

If f(x) = xn, for some natural number n (including zero) then

f'(x) = nx^{n-1}.\,

Special cases include:

  • Constant rule: if f is the constant function f(x) = c, for any real number c, then
f' = 0 \,
  • The derivative of a linear function is constant: if f(x) = ax (or more generally, in view of the constant rule, if f(x)=ax+b), then
f'(x) = a.\,

Combining this rule with the linearity of the derivative permits the computation of the derivative of any polynomial.

[edit] The reciprocal rule

Main article: Reciprocal rule

For any (nonvanishing) function f, the derivative of the function 1/f (equal at x to 1/f(x)) is

-\frac{f'}{f^2}.\,

In other words, the derivative of h(x) = 1/f(x) is

h'(x) = -f '(x)/f(x)2.

In Leibnitz's notation, this is written

\frac{d(1/f)}{dx} = -\frac{1}{f^2}\frac{df}{dx}.\,

[edit] The inverse function rule

This should not be confused with the reciprocal rule: the reciprocal 1/x of a nonzero real number x is its inverse with respect to multiplication, whereas the inverse of a function is its inverse with respect to function composition.

If the function f has an inverse g = f-1 (so that g(f(x)) = x and f(g(y)) = y) then

g' = \frac{1}{f'\circ f^{-1}}.\,

In other words, if y = f(x) has an inverse x = g(y), then

g'(y) = 1/f '(f-1(y)).

In Leibniz notation, this is written (suggestively) as

\frac{dx}{dy} = \frac{1}{dy/dx}.

[edit] Further rules of differentiation

[edit] The quotient rule

Main article: Quotient rule

If f and g are functions, then:

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}\quad wherever g is nonzero.

This can be derived from reciprocal rule and the product rule. Conversely (using the constant rule) the reciprocal rule is the special case f(x) = 1.

[edit] Generalized power rules

The elementary power rule generalizes considerably. First, if x is positive, it holds when n is any real number. The reciprocal rule is then the special case n = -1 (although care must then be taken to avoid confusion with the inverse rule).

The most general power rule is the functional power rule: for any functions f and g,

(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\quad

wherever both sides are well defined.

[edit] Logarithmic derivatives

The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):

f' = (\ln f)'f\quad wherever f is positive.

[edit] See also

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