# What are differentiable points for a function?

##### 1 Answer
Dec 3, 2015

A differentiable point of a function $f \left(x\right)$ is a value $a$ such that the two-sided limit exists and is finite:

${\lim}_{h \to 0} \frac{f \left(a + h\right) - f \left(a\right)}{h}$

#### Explanation:

For example, let us verify that $f \left(x\right) = {x}^{2} + x$ is differentiable at $x = 2$:

Let $a = 2$

Then:

${\lim}_{h \to 0} \frac{f \left(a + h\right) - f \left(a\right)}{h}$

$= {\lim}_{h \to 0} \frac{f \left(2 + h\right) - f \left(2\right)}{h}$

$= {\lim}_{h \to 0} \frac{\left({\left(2 + h\right)}^{2} + \left(2 + h\right)\right) - \left({2}^{2} + 2\right)}{h}$

$= {\lim}_{h \to 0} \frac{\left(4 + 4 h + {h}^{2} + 2 + h\right) - \left(4 + 2\right)}{h}$

$= {\lim}_{h \to 0} \frac{5 h + {h}^{2}}{h}$

$= {\lim}_{h \to 0} \left(5 + h\right)$

$= 5$