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DIFFERENTIABILITY
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DIFFERENTIABILITY
INTRODUCTION

5)
Proof:
Let f(x) = sinx then,

Hence,

6)
7)
8)
9)
10)  Why?
Proof:
Let f(x) = secx

Hence,
11)
Proof:
Let  then x = siny.
Differentiating w.r.t x we get,

Hence,
12)
13)
14)
15)
16) Why?
Proof:
Let y = sec-1x then secy = x.
Differentiating w.r.t x we have,

Hence,

Dumb Question:
Why and not ?
Ans:

Fig (4)
Note that y = sec-1x is an increasing function in its domain.
So,
Illustration 6:
Differentiate
Solution:

Different Methods of Differentiation:
1) Differentiation of a function defined parametrically:
Let x, y be function of parameter t, i.e.
x = f (t), y = f (t) then,

Illustration:

Solution:

2) Logarithmic Differentiation:
The process of taking logarithm on both sides and then differentiating is called logarithmic differentiation.
Illustration 8:
Differentiate w.r.t x.
Solution:
Let y = .
Taking logarithm on both sides, then we have
logy = sinx.logx

3) Differentiation of Implicit function:
If the relation between x and y is given by equation containing both and this equation is not immediately solvable for y, then y is called implicit function of x.
Illustration 9:
If  Then prove that
Solution:
Given that  differentiating on both sides w.r.t x we have,

Hence,

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