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

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)
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

Set all terms to a common denominator

Expand and group to obtain the derivative f'.
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

Set a common denominator to all terms

Expand and group like terms to obtain the derivative f'.
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

Calculate U ' and substitute above to obtain 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

Calculate U ', using the quotient rule, and substitute to obtain

Expand and group like terms to obatin a final form for the derivative f '
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

Calculate U ', substitute and simplify to obtain the derivative f '.
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

Since U is the quotient of two function, use the quotient rule to find U ' and substitute to obtain

Expand and group like terms

Change the negative exponent into a positive exponent to find a final form of 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 