Find Derivatives of Functions in Calculus




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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

find derivative function example 1

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:

    derivative solution to example 1, step 1
  • Expand and group to obtain

    derivative solution to example 1, step 2



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

find derivative function example 2

Solution to Example 2:

  • This function may be considered as the product of function U = sqrt x + 2x and V = 4x2 - 1, hence the use of the product rule

    derivative solution to example 2, step 1
  • To add the above you need to write all terms as fractions with a common denominator.

    derivative solution to example 2, step 2
  • Expand

    derivative solution to example 2, step 3
  • and group to obtain the final result for the derivative of f as follows.

    derivative solution to example 2, step 4


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

find derivative function example 3

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

    derivative solution to example 3, step 1
  • 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

find derivative function example 4

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

find derivative function example 5

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 considedered 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

find derivative function example 6

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

find derivative function example 7

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

find derivative function example 8

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

find derivative function example 9

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

find derivative function example 10

Solution to Example 10:

  • The given function is of the form U3/2 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

find derivative function example 11

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

    derivative solution to example 11, step 1
  • Since U is the quotient of two function, use the quotient rule to find U ' and substitute to obtain

    derivative solution to example 11, step 2
  • Expand and group like terms

    derivative solution to example 11, step 3
  • Change the negative exponent into a positive exponent to find a final form of f ' as follows

    derivative solution to example 11, step 4


Exercises: Find the derivative of each of the following functions.

Find derivatives of functions

Answers to Above Exercises:

answers to above questions


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Updated: 2 April 2013

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