Techniques for differentiation

From Wikipedia, the free encyclopedia

This article contains a list of techniques for the differentiation of real functions, categorized by type.

1 Simple polynomial functions
2 Exponential functions
3 Logarithmic functions
4 Trigonometric functions

[edit] Simple polynomial functions

Given a polynomial $p (x)$ , that is defined by the formula:

$p(x) = \sum^m_{i=0} k_i x^i$ , one has

$\frac{d}{dx} p(x) = \sum^m_{i=0} ik_ix^{i-1}.$

That is, one simply multiplied each term by its degree, then divides by ‘’x’’. For example, one can differentiate $\sqrt{x} + 5x$ . First, one would break it down into its component terms: sqrt(x) and 5x. sqrt(x) is equal to x^1/2, meaning that its derivative is 1/(2sqrt(x)), or half the reciprocal of the value. 5x simply becomes 5, giving us:

$\frac{d}{dx} (\sqrt{x} + 5x) = \frac{1 + 10\sqrt{x}}{2\sqrt{x}}.$

[edit] Exponential functions

Given some function ‘’f(x)’’ equals b^x, its derivative can be found via the following formula:

$\frac{d}{dx} b^x = b^x \ln b$

where ‘’ln b’’ is the natural logarithm of b. Using this formula, we can differentiate 225^x, which gives us 2(ln 3 + ln 5). (See Natural logarithm). So ultimately, we have b^x 2ln 2 + b^x 2ln 5.

[edit] Logarithmic functions

All logarithmic functions can be differentiated via a formula very similar to that for exponential functions. The slope of any logarithmic function at a point x is equal to the reciprocal of x times the natural logarithm of the base, or:

$\frac{d}{dx} \log_b x = \frac{1}{x \ln b}.$

Through this we can differentiate the natural logarithm itself. Of course, the base of the natural logarithm is e, and the base x logarithm of x is always one. Therefore, the natural logarithm of e is one. Knowing this, we can find that the slope of the natural logarithm at any point equals the reciprocal of the height at that point.