Free Mathematics Tutorials

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 = x² - 5 and V = x³ - 2 x + 3; hence function example 1 written as a product of two functions

We use the product rule to differentiate f as follows: derivative of a product of two functions

where U ' and V ' are the derivatives of U and V respectively and are given by derivatives of functions U and V

Substitute to obtain derivatives of functions f

Expand, group and simplify to obtain derivatives of functions f simplified

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

f(x) = (\sqrt x + 2x)(4x^2-1)

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

f'(x) = U' V + U V' \\ = (\dfrac{1}{2\sqrt x} + 2)(4x^2-1) + (\sqrt x + 2 x)(8x)

To add the above you need to write all terms as fractions with a common denominator.

f'(x) = \dfrac{(1+2\cdot2\sqrt x)(4x^2-1)+2\sqrt x(8x)(\sqrt x + 2x)}{2\sqrt x}

Expand

f'(x) = \dfrac{4x^2-1+16x^{5/2}-4\sqrt x+16x^2+32x^{5/2}}{2\sqrt x}

and group to obtain the final result for the derivative of f as follows.

f'(x) = \dfrac{48x^{5/2}+20x^2-4x^{1/2}-1}{2\sqrt x}

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

Expand and group to obtain f'(x) as follows
derivative solution to example 3, step 2

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
derivative solution to example 4, step 1

Write all terms in the numerator so that they have the same denominator 2 sqrt x
derivative solution to example 4, step 2

Expand and group like terms to obtain f'(x)
derivative solution to example 4, step 3

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
derivative solution to example 5, step 1

Set all terms to a common denominator
derivative solution to example 5, step 2

Expand and group to obtain the derivative f'.
derivative solution to example 5, step 3

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
derivative solution to example 6, step 1

Set a common denominator to all terms
derivative solution to example 6, step 2

Expand and group like terms to obtain the derivative f'.
derivative solution to example 6, step 3

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
derivative solution to example 7, step 1

Calculate U ' and substitute above to obtain f ' as follows
derivative solution to example 7, step 2

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
derivative solution to example 8, step 1

Calculate U ', using the quotient rule, and substitute to obtain
derivative solution to example 8, step 2

Expand and group like terms to obatin a final form for the derivative f '
derivative solution to example 8, step 3

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
derivative solution to example 9

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 = x² + 5. Apply the chain rule as follows
derivative solution to example 10, step 1

Calculate U ', substitute and simplify to obtain the derivative f '.
derivative solution to example 10, step 2

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
derivative solution to example 11, step 1