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Solved Examples on Differentiability


Illustration 1: Let [.] denotes the greatest integer function and f(x) = [tan2x], then does the limit exist or is the function differentiable or continuous at 0? (1999)

Solution: Given f(x) = [tan2x]

Now, -45°< x < 45°

tan(-45°)< tanx < tan45°

-tan 45°< tan x < tan 45°

-1< tan x <1

So, 0 <tan2x < 1

[tan2x] = 0

So, f(x) is zero for all values of x form x = -45° to 45°.

Hence, f is continuous at x =0 and f is also differentiable at 0 and has a value zero.



Illustration 2: A function is defined as follows:

f(x)= x3  , x2< 1

  x   , x2≥ 1

Discuss the differentiability of the function at x=1.

Solution:We have R.H.D. = Rf'(1)

                                    = limh→0 (f(1-h)-f(1))/h

= limh→0 (1+h-1)/h = 1

and L.H.D. = Lf'(1)= limh→0 (f(1-h)-f(1))/(-h)

                 = limh→0 ((1-h)3-1)/(-h)

= limh→0 (3-3h+h2) = 3

?Rf'(1)≠ Lf'(1)⇒ f(x) is not differentiable at x=1.

Illustration 3:If y = (sin-1x)2 + k sin-1x, show that (1-x2) (d2 y)/dx2 - x dy/dx = 2

Solution: Here y = (sin-1x)2 + k sin-1x.

Differentiating both sides with respect to x, we have

dy/dx = 2(sin-1 x)/√(1-x2 ) + k/√(1-x2 )

⇒(1-x2 ) (dy/dx)2 = 4y + k2

Differentiating this with respect to x, we get

(1-x2) 2 dy/dx.(d2 y)/(dx2 ) - 2x (dy/dx)2 = 4(dy/dx)
⇒(1-x2 ) ( d2 y)/dx2 -x dy/dx = 2

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