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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