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Derivatives - Review

This review page contains a summary of differentiation rules, that is, of rules for computing the derivative of a function. If $fleft( x
ight) $ is a function, its first derivative is denoted by [eq1].

Table of Contents

Derivative of a constant function

If $fleft( x
ight) $ is a constant function[eq2]where $cin U{211d} $, then its first derivative is[eq3]

Derivative of a power function

If $fleft( x
ight) $ is a power function[eq4]then its first derivative is[eq5]where $nin U{211d} $ is a constant.

Derivative of a logarithmic function

If $fleft( x
ight) $ is the natural logarithm of x, that is,[eq6]then its first derivative is[eq7]

If $fleft( x
ight) $ is the logarithm to base $b$ of x, that is,[eq8]then its first derivative is[eq9](remember that [eq10]).

Derivative of an exponential function

If $fleft( x
ight) $ is the exponential function[eq11]then its first derivative is[eq12]

If the exponential function $fleft( x
ight) $ does not have the natural base $e$, but another positive base $b$, that is, if[eq13]then its first derivative is[eq14](remember that [eq15]).

Derivative of a linear combination of functions

If [eq16] and [eq17] are two functions and [eq18] are two constants, then[eq19]

In other words, the derivative of a linear combination is equal to the linear combinations of the derivatives. This property is called "linearity of the derivative".

Two special cases of this rule are[eq20]

Derivative of a product of functions

If [eq16] and [eq17] are two functions, then the derivative of their product is[eq23]

Derivative of a composition of functions (chain rule)

If $fleft( x
ight) $ and $gleft( y
ight) $ are two functions, then the derivative of their composition is[eq24]

What does this chain rule mean in practice? It means that first you need to compute the derivative of $gleft( y
ight) $:[eq25]Then, you substitute $y$ with $fleft( x
ight) $:[eq26]Finally, you multiply it by the derivative of $fleft( x
ight) $:[eq27]

Derivatives of trigonometric functions

The trigonometric functions have the following derivatives:[eq28]while the inverse trigonometric functions have the following derivatives:[eq29]

Derivative of an inverse function

If [eq30] is a function with derivative[eq31]then its inverse [eq32] has derivative[eq33]

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