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:

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. _aa^½

_aa

_ae_ae

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. log_a
2. log_e
3. log_e

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. ^x½

Solution:

1. e^tanx

Let ^{tanx u}

^u ; ² ⇒^u.^{2 2tanx}

1. ^cosx

Let ^cosx then, _ea.

_ea ⇒ _ea ^cosxlog_ea.

1. 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

1. a^2x – a^-2x
2. 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:

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 -½ 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
• GRADIENT