# WEEK TWO SS2 FURTHER MATHS SECOND TERM DIFFERENTIATION 2

**WEEK TWO **

**SS2 FURTHER MATHS SECOND TERM**

**DIFFERENTIATION 2**

**CONTENT**

(a) Differentiation of Transcendental Functions such as Sin x,e^{ax} log 3x.

(b) Rules of Differentiation: Product rule, Quotient rule, Function of function.

(c) Higher Derivative

**SUB TOPIC: DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS**

The following are called **transcendental** functions and their derivatives are called **identities**:

- Derivative of sin: If
- Derivative of
- Derivative of
^{2}. Recall that - Derivative of sec Recall that sec
- Derivative of cosec if
- The derivative of
^{2}. Recall that

**Examples:**

- Differentiate
^{2}

Solution:

^{22}. If , then ^{2}

- Differentiate the following with respect to
- b.
^{2}c.

Solution:

^{2}^{22})^{222}.^{222}

**CLASS ACTIVITY**:

- Find the derivative of each of the following:
^{2}- Show that
^{2}

**LOGARITHMIC FUNCTION**

Given that _{e} then,

**Examples**:

Find the derivative with respect to of each of the following:

_{a }b._{10}(^{2}c._{e}

Solution:

_{aa}^{½}

_{aa}

_{a}e_{a}e

_{10}(^{2}let u = (^{2}and_{10}u.

_{e}10; (^{2}_{10}e

_{ee}u.

^{2} and

^{2}) ^{2})

**CLASS ACTIVITY**:

Find the derivative with respect to of each of the following:

- log
_{a } - log
_{e } - log
_{e}

**DERIVATIVE OF EXPONENTIAL FUNCTIONS**

Given ^{x} where

Then ^{x}. Also ^{x }⇒ _{e}*a*

**Examples**:

Differentiate with respect the following:

^{tanx}b.^{cosx}c.^{x-x}d.^{x½}

Solution:

- e
^{tanx}

Let ^{tanx} ^{u}

^{u} ; ^{2 } ⇒^{u}.^{2} ^{2tanx}

^{cosx}

Let ^{cosx} then, _{e}a.

_{e}a ⇒ _{e}a ^{cosx}log_{e}a.

- e
^{x}– e^{-x}

e^{x }– e^{-x}

e^{x }– ( – e^{x}) = e^{x }+ e^{-x}

**CLASS ACTIVITY:**

Differentiate each of the following with respect to[mediator_tech]

1.a^{1+tanx}

- a
^{2x}– a^{-2x} - a
^{tanx}

**SUB TOPIC: RULES OF DIFFERENTIATION**

**DERIVATIVE OF A SUM**

Supposing and are functions of such that: ;

then,

Hence, the derivative of a sum is the sum of the derivatives.

Similarly, the derivative of a different is the different of the derivative.

That is:

**Examples**:

- find the differential coefficient of
^{63}

solution:

^{636352}

- Differentiate with respect to ,
^{2}

Solution:

^{22}

**FUNCTION OF A FUNCTION**

Given that is a function of and itself is also a function of , then the derivative of with respect to can be find using this method:

this method is also called **CHAIN RULE**.

**Example:**

- Differentiate
^{2}

Solution:

Let ^{2 } ⇒^{½ }

^{½ – 1} ^{-½} ^{2}

^{22}

Let U

**DERIVATIVE OF A PRODUCT OF FUNCTION**.

If where are both functions of then

**Example:**

- Differentiate with respect to
^{34}

Solution:

^{34}

^{2} ^{3}

^{33} ^{4}. ^{2632}

- Find with respect to
^{2}

**DERIVATIVE OF A QUOTIENT FUNCTION**

If are both functions of then:

*V ^{2}*

This formula is called ** quotient rule**.

**Example**:

Find the derivative of

Solution:

Let

*(x+1) ^{2}*

Simplify the above to the minimum value.

**CLASS ACTIVITY:**

- Differentiate
^{24}with respect to using chain rule. - Given that
^{4 }, find . - Differentiate
- Differentiate
- Differentiate with respect to
^{2}+5)^{44})

**SUB TOPIC: HIGHER DERIVATIVES**

This implies finding the derivative of a given function beyond first derivative. It can extend to any number of derivatives as so required.

**Example**:

Find of the following:

^{5}b.^{63-3x}

Solution:

^{5}

**CLASS ACTIVITY:**

^{432}

- Given that

Find

- Use quotient rule to find the derivative of each of the following:
- b.

**KEY WORDS**

**LIMIT****CONTINUITY****FUNCTION****DERIVATIVE****DIFFERENTIATION****FIRST PRINCIPLE****WITH RESPECT TO****POLYNOMIALS****GRADIENT**