**
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