Examples
Ex1:
Let f: (–1, 1) → R be a differentiable function with f(0) = –1 and f '(0) = 1.
Let g(x) = [f(2f(x) + 2)]2. Then find g'(0).
Sol:
Given g(x) = [f(2 f(x) + 2)]2
⇒ g'(x) = 2 f(2 f(x) + 2)).f '(2 f(x) + 2).2f '(x)
Substituting x = 0,
⇒ g'(0) = 2f(2 f(0) + 2)).f '(2f(0) + 2).2f '(0)
= 2f(0).f '(0).2f '(0)   (∵ f(0) = –1 and f '(0) = 1)
= 2(–1)(1).2(1) = – 4
Ex2:
Show that the function y = e–|x| is continuous and not differentiable at x = 0.
let f(x) =
Sol:
Clearly, f(x) is continuous and differentiable for all non-zero x.

Now f(x) = ex = 1
f(x) = e–x = 1
Also f(0) = e0 = 1
So, f(x) is continuous for all x.
So, f(x) is not differentiable at x = 0.
Hence f(x) = e–|x| is continuous everywhere but not differentiable at x = 0.
Ex3:
Let f(x) =
f(x) is continuous but not differentiable at x = 0. Find 'p'.
Sol:
Given f(x) =
At x = 0, f(x) is a continuous function
∴ f(x) = f(0) = 0
⇒ xp = 0
⇒ p > 0
f(x) is differentiable at x = 0, if exists.
⇒ exists
⇒ exists
⇒ p – 1 > 0 or p > 1
If p ≤ 1, then xp – 1 does not exist
and f(x) is not differentiable at x = 0.
∴ For 0 < p ≤ 1, f(x) is a continuous function at x=0 but not differentiable.