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)

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

= lim_h→0 (1+h-1)/h = 1

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

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

= lim_h→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