Differentiation Knowledge Application Example

Lets assume that the student asked the solutions to the below questions.

 

Question: Differentiate the following functions.

a)                       b)

c)                                                    d)

e)                                  f)

 

Answer:

 

a)   The question (a) uses the derivative rules (Differentiation Download)

1.      Derivative of a constant

2.      Power rule

3.      Product of a constant and a function

4.      Sum of functions

5.      Difference of functions

      You can solve this question by seeing the solution to the given question (a), which
      use the same concepts.

 

b)   The question (b) uses the same derivative rules as question (a). It also uses index rules
      (Indices and Logarithms Download). You can solve this question by seeing the
      solution to the given question (b), which use the same concepts.

 

c)   The question (c) uses the same derivative rules as question (b). You can solve this
      question by seeing the solution to the given question (c), which uses the same
      concepts.

 

d)   The question (d) uses the chain rule. You can solve this question by seeing the
      solution to the given question (d) which also uses the chain rule.

 

e)   The question (e) uses the product rule. You can solve this question by seeing the
      solution to the given question (e) which also uses the product rule.

 

f)   The question (f) uses the quotient rule. You can solve this question by seeing the
      solution to the given question (f) which also uses the quotient rule.

 

 

 

Derivative Rules and Theorems

1.   Derivative of a constant is zero.

2.  Power rule: If  where k and n are constants, then

3.  Product of a constant and a function:  

4.  Sum of functions:

5.   Difference of functions:

6.   Composite functions (chain rule):  If  and  then

7.   Product rule:  If y, u and v are functions of x and  then

8.   Quotient rule: If y, u and v are functions of x and  then  

Note that:  can be written as    and then the product rule may be used.

 

 

Questions which will help you to solve your questions.

 

a)   Differentiate

 

Answer: The variable is s and therefore the derivative is taken with respect to s.

 

         (Remember, the derivative of a

                                                                                  constant is zero.)

     

 

b)   Differentiate

 

Answer: The variable is u and therefore the derivative is taken with respect to u.

 

                  (Remember the index rule )

     

     

                         (Remember the index rule )

 

 

c)   Differentiate

 

Answer: The variable is m and therefore the derivative is taken with respect to m.

 

     

                       (Remember the index rule  and )

     

     

     

 

d)   Differentiate

 

Answer: The variable is m and therefore the derivative is taken with respect to m. The
      differentiation is done by using the chain rule. That is .

     

     

       

e)   Differentiate

 

Answer: The variable is x and therefore the derivative is taken with respect to x.

 

     

      Where                   and         

                                                         

     

      Then         

      Hence       

                       

                       

                       

 

      You could also expand f(x) and then take the derivative. However, when the
      bracketed terms have indices  then the expansion is no longer
      practical.

 

f)   Differentiate

 

Answer: The variable is t and therefore the derivative is taken with respect to t. The
      function is a quotient and therefore quotient rule is used.

     

     

 

      Where                and        

                                                    

     

      Then         

      Hence