# WEEK THREE SS2 FURTHER MATHS SECOND TERM DIFFERENTIATION 3

WEEK THREE

SS2 FURTHER MATHS SECOND TERM

DIFFERENTIATION 3

CONTENT

(a) Application of Differentiation

(b) Differentiation of Implicit Functions.

SUB TOPIC: APPLICATION OF DIFFERENTIATION

Examples:

1. Suppose an object is dropped from rest from a given height, the distance s through which the object has dropped after t seconds (ignoring air resistance) is given by the formula

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1. Let p(t) denote the population of a colony of bacteria after t hours. If how fast is the population growing after 3hours?

Solution:

1. We want the instantaneous rate of growth of growth of the population when t = 3hours.

CLASS ACTIVITY:

1. The distance (d metres) fallen by a stone in t seconds is given by the equation
2. Find the maximum and minimum points of the curve and sketch the curve.

SUB TOPIC: IMPLICIT DIFFERENTIATION

Functions which have their derivatives in terms of it may be possible to separate completely from to different side. Such functions are called implicit function.

Examples are 22

Example:

1. If 3233 find

Solution:

2

22

Solving for

) 2

.

1. If 4342 find

Solution:

Differentiating both side with respect to

33342)

342)

Solving for

33

CLASS ACTIVITY:

Differentiate the following:

1. 2233
2. 32+
3. 2

PRACTICE EXERCISE:

Essay:

1. If
2. Given that
3. 2
4. A ball rolls down an inclined plane. It travels s centimeters in t seconds and (i) how far has the ball travelled in the first second? (ii) what is its speed at the end of the fourth second?
5. A particle moves along a straight so that after a time t seconds, its distance Scm from the starting point O is given by find: (a) the distances from O when the particle is momentarily at rest; (b) the velocity when the acceleration is zero.

ASSIGNMENT

1. Differentiate with respect to x:
2. Find from first principle, the derivative, with respect to x of the function
3. Find
4. Find
5. If calculate the average rate of change(gradient) of y, w.r.t x, in the interval between x = c and x = c + h. hence find . Also find the maximum and minimum points of the curve defined by and sketch the curve.

KEY WORDS

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