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   