NDA Math Differentiation

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)≠

lim

x→a−

f′(x),
the given function is not differentiable at x=0, or, f′(0) does not exist.