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)

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.

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