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What are differentiable points for a function?

1 Answer

Dec 3, 2015

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

$lim_(h->0) (f(a+h)-f(a))/h$

Explanation:

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

Let $a = 2$

Then:

$lim_(h->0) (f(a+h)-f(a))/h$

$=lim_(h->0) (f(2+h)-f(2))/h$

$=lim_(h->0) (((2+h)^2+(2+h))-(2^2+2))/h$

$=lim_(h->0) ((4+4h+h^2+2+h)-(4+2))/h$

$=lim_(h->0) (5h+h^2)/h$

$=lim_(h->0) (5+h)$

$=5$

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