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# Find Derivatives of Functions in Calculus

Find the derivatives of various functions using different methods and rules. 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 -2x+3. We use the product rule to differentiate f as follows:

Expand and group to obtain

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 = sqrt x + 2x and V = 4x 2 - 1, hence the use of the product rule

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

Expand

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

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

Expand and group to obtain f'(x) 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

Write all terms in the numerator so that they have the same denominator 2 sqrt x

Expand and group like terms to obtain f'(x)

Source: www.analyzemath.com
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