Derivatives - Review

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

After reviewing the differentiation rules, you can take multiple choice tests to assess your knowledge: Test 1, Test 2, Test 3, Test 4.

Derivative of a constant function

If is a constant function:where , then its first derivative is:

Derivative of a power function

If is a power function:then its first derivative is:where is a constant.

Derivative of a logarithmic function

If is the natural logarithm of :then its first derivative is:

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

Derivative of an exponential function

If is the exponential function:then its first derivative is:

If the exponential function does not have the natural base , but another positive base :then 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 derivative:then its inverse has derivative: