Concept: - Differentiability of a Function: A function f(x) is differentiable at x=a in its domain if its derivative is continuous at a. This means that f(a) must exist, or equivalently:
lim
x→a+
f′(x)=
lim
x→a−
f′(x)=
lim
x→a
f′(x)=f′(a). - The Modulus Function ' ∣ ' is defined as: |x|={x,x≥0 −x,x<0 Calculation: Using the definition of Modulus Function, we have: f(x)=ex,x≥0 And, f(x)=e−x,x<0. Using the first principle of derivatives, we find that:
lim
x→0+
f′(x)=
lim
x→0
ex=1 And,
lim
x→0−
f′(x)=
lim
x→0
−e−x=−1. Since
lim
x→a+
f′(x)â‰
lim
x→a−
f′(x), the given function is not differentiable at x=0, or, f′(0) does not exist.