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Prove that $f(x)$ is differentiable at $x$ if and only if $\lim_{t\to x}\,\frac{f(t)-f(x)}{t-x}$ exists. When the limit exists it is equal to $f'(x)$ .

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Lesson Parent:

Differentiation II: The Gory Details of Calculating Derivatives

Problem:

Prove that $f(x)$ is differentiable at $x$ if and only if
$\lim_{t\to x}\,\frac{f(t)-f(x)}{t-x}$
exists. When the limit exists it is equal to $f'(x)$ .

Answer:

$\frac{df}{dx}=\lim_{t\to x}\frac{f(t)-f(x)}{t-x}$

Solution:

Let's define

<Sign in to see all the formulas>

where the domain of

$e(t)$ is the domain of

$f(x)$ except that

$e(t)$ is not defined at

$t=x$ . Now note that

<Sign in to see all the formulas>

and that

$e(t)$ not being defined at

$t=x$ is equivalent to <Sign in to see all the formulas> not being defined at

$h=0$ .

Recall that when we say "if $A$ then $B$ " we symbolically denote this by <Sign in to see all the formulas>. When we say " $A$ if and only if $B$ " we symbolically denote this by <Sign in to see all the formulas>. To prove statements like " $A$ if and only if $B$ " we need to show that