# 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,eax 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:

1. Derivative of sin: If
2. Derivative of
3. Derivative of 2. Recall that
4. Derivative of sec Recall that sec
5. Derivative of cosec if
6. The derivative of 2. Recall that

Examples:

1. Differentiate 2

Solution:

22. If , then 2

1. Differentiate the following with respect to
2. b. 2 c.

Solution:

1. 2 22)222.
2. 222

CLASS ACTIVITY:

1. Find the derivative of each of the following:
2. 2
3. Show that 2

LOGARITHMIC FUNCTION

Given that e then,

Examples:

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

1. a b. 10(2 c. e

Solution:

1. aa½

aa

aeae

1. 10(2 let u = (2 and 10u.

e10; (210e

1. eeu.

2 and

2) 2)

CLASS ACTIVITY:

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

1. loga
2. loge
3. loge

DERIVATIVE OF EXPONENTIAL FUNCTIONS

Given x where

Then x. Also x ea

Examples:

Differentiate with respect the following:

1. tanx b. cosx c. x-x d.

Solution:

1. etanx

Let tanx u

u ; 2 u.2 2tanx

1. cosx

Let cosx then, ea.

ea ⇒ ea cosxlogea.

1. ex – e-x

ex – e-x

ex – ( – ex) = ex + e-x

CLASS ACTIVITY:

Differentiate each of the following with respect to[mediator_tech]

1.a1+tanx

1. a2x – a-2x
2. atanx

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:

1. find the differential coefficient of 63

solution:

636352

1. 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:

1. Differentiate 2

Solution:

Let 2 ½

½ – 1 2

22

Let U

DERIVATIVE OF A PRODUCT OF FUNCTION.

If where are both functions of then

Example:

1. Differentiate with respect to 34

Solution:

34

2 3

33 4. 2632

1. Find with respect to 2

DERIVATIVE OF A QUOTIENT FUNCTION

If are both functions of then:

V2

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:

1. Differentiate 24 with respect to using chain rule.
2. Given that 4 , find .
3. Differentiate
4. Differentiate
5. Differentiate with respect to
6. 2+5)
7. 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:

1. 5 b. 63-3x

Solution:

5

CLASS ACTIVITY:

432

1. Given that

Find

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

KEY WORDS

• LIMIT
• CONTINUITY
• FUNCTION
• DERIVATIVE
• DIFFERENTIATION
• FIRST PRINCIPLE
• WITH RESPECT TO
• POLYNOMIALS