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

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 = 4x2 - 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 = x2 + 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 = (x2 + 3)/(2x - 1), and function V may be considered as the quotient of two functions x2 + 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)(x3 - x) and also consider V as the product of (2x - 1) and (x3 - 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 U4 where U = x3 + 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 U3 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 = x3 + 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 U3/2 with U = x2 + 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 U1/4 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