# Find Derivatives of Functions in Calculus

Find the derivatives of various functions using different methods and rules in calculus. Several Examples with detailed solutions are presented. More exercises with answers are at the end of this page.

** Example 1:** Find the derivative of function f given by

__Solution to Example 1:__Function f is the product of two functions: U = x

^{ 2}- 5 and V = x

^{ 3}- 2 x + 3; hence

** Example 2:** Calculate the first derivative of function f given by

__Solution to Example 2:__This function may be considered as the product of function U = √x + 2x and V = 4x

^{2}- 1, hence the use of the product rule

** Example 3:** Calculate the first derivative of function f given by

__Solution to Example 3:__

The given function may be considered as the ratio of two functions: U = x^{2} + 1 and V = 5x - 3 and use the quotient rule to differentiate f is used as follows

** Example 4:** Calculate the first derivative of function f given by

__Solution to Example 4:__Function f is the quotient of two functions hence the use of the quotient rule

** Example 5:** Calculate the first derivative of function f given by

__Solution to Example 5:__Function f given above may be considered as the product of functions U = 1/x - 3 and V = (x

^{2}+ 3)/(2x - 1), and function V may be considedered as the quotient of two functions x

^{2}+ 3 and 2x - 1. We use the product rule for f and the quotient rule for V as follows

** Example 6:** Calculate the first derivative of function f given by

__Solution to Example 6:__There are several ways to find the derivative of function f given above. One of them is to consider function f as the product of function U = sqrt x and V = (2x - 1)(x

^{3}- x) and also consider V as the product of (2x - 1) and (x

^{3}- x) and apply the product rule to f and V as follows

** Example 7:** Find the derivative of function f given by

__Solution to Example 7:__The given function is of the form U

^{4}where U = x

^{3}+ 4. The use of the chain rule of differentiation gives f ' as follows

** Example 8:** Find the derivative of function f given by

__Solution to Example 8:__Function f is of the form U

^{3}where U = (x - 1) / (x + 3). Apply the chain rule to obtain f ' as follows

** Example 9:** Find the derivative of function f given by

__Solution to Example 9:__The given function is of the form sqrt U with U = x

^{3}+ 2 x + 1. Calculate U ' and use the chain rule to obtain

** Example 10:** Find the derivative of function f given by

__Solution to Example 10:__The given function is of the form U

^{3/2}with U = x

^{2}+ 5. Apply the chain rule as follows

** Example 11:** Find the derivative of function f given by

__Solution to Example 11:__Function f is of the form U

^{1/4}with U = (x + 6)/(x + 5). Use the chain rule to calculate f ' as follows

__Exercises:__ Find the derivative of each of the following functions.

__Answers to Above Exercises:__

More on

differentiation and derivatives

and also

Find Derivatives of Rational Functions - Calculators