Introduction
A function is differentiable if it has a finite derivative at every point of its domain (Its input set).
Below are some functions and their domains:
\begin{align*}
&f(x)=x^2 &\quad& \mathbb{R} \\
&f(x)=\sin(x) &\quad& \mathbb{R} \\
&f(x)=\sqrt{x} &\quad& \lbrack0, \infty \rparen \\
&f(x)=\frac{1}{x} &\quad& (-\infty, 0) \cup (0, \infty)
\end{align*}
A function is only differentiable if
1. It is continuous.
(If f
has a hole or jump, it does not have a finite rate of change.)
2. It is smooth. (It has no cups or corners.)
(A derivative is undefined at corners and cusps, where the left and right sides of the derivative's limit do not match.)
3. It has no vertical tangents.
(A derivative is infinite at a vertical tangent.)Examples
All the examples below are not differentiable.
Hole at
x=1
f(x)=\frac{x^2-1}{x-1}
Jump at
x=0
f(x)=\frac{x}{|x|}
Corner at
x=0
(Corner: The left and right side of the derivative's limit do not match and are finite.)
f(x)=|x|
Cusp at
x=0
(Cusp: The left and right side of the derivative's limit do not match and are infinite.)
f(x)=x^{2/3}
Vertical tangent at
x=0
(Vertical tangent: The derivative is infinite.)
f(x)=\sqrt[3]{x^2}