Derivatives - Review

by Marco Taboga, PhD

This review page contains a summary of differentiation rules, that is, of rules for computing the derivative of a function. If is a function, its first derivative is denoted by .

Derivative of a constant function

If is a constant functionwhere , then its first derivative is

Derivative of a power function

If is a power functionthen its first derivative iswhere is a constant.

Derivative of a logarithmic function

If is the natural logarithm of , that is,then its first derivative is

If is the logarithm to base of , that is,then its first derivative is(remember that ).

Derivative of an exponential function

If is the exponential functionthen its first derivative is

If the exponential function does not have the natural base , but another positive base , that is, ifthen its first derivative is(remember that ).

Derivative of a linear combination of functions

If and are two functions and are two constants, then

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

Derivative of a product of functions

If and are two functions, then the derivative of their product is

Derivative of a composition of functions (chain rule)

If and are two functions, then the derivative of their composition is

What does this chain rule mean in practice? It means that first you need to compute the derivative of :Then, you substitute with :Finally, you multiply it by the derivative of :

Derivatives of trigonometric functions

The trigonometric functions have the following derivatives:while the inverse trigonometric functions have the following derivatives:

Derivative of an inverse function

If is a function with derivativethen its inverse has derivative

How to cite

Please cite as:

Taboga, Marco (2021). "Derivatives - Review", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/mathematical-tools/derivatives-review.