Equivalence of Definitions of Derivative

Theorem
The two forms of the definition of a derivative of a real function at any point $(c, f(c))$ are consistent. That is, for any constant $c$ in the domain of $f$ for which $f^\prime\left(c \right)$ exists,


 * $\displaystyle f^\prime \left({c}\right) = \lim_{\Delta x \to 0} \frac {f \left({c + {\Delta x}}\right) - f \left({c}\right)} {\Delta x}$

and


 * $\displaystyle f^\prime \left({c}\right) = \lim_{x \to c} \frac {f \left({x}\right) - f \left({c}\right)} {x - c}$

Are equivalent.

Informal Proof
$\Delta x$ represents the distance on the graph between any point $(x,f(x))$ and $(c,f(c))$. As the two points approach each other, the distance $\Delta x$ between them also gets smaller. When the two points are infinitely close together, $\Delta x$ is infinitely small, so the two limits are the same.

Part 1

 * $\displaystyle f^\prime \left({c}\right) = \lim_{\Delta x \to 0} \frac {f \left({c + {\Delta x}}\right) - f \left({c}\right)} {\Delta x}$.

Define $x$ as


 * $x = c + \Delta x$.

$\implies$


 * $x - c = \Delta x$.

Then $x \to c \iff \Delta x \to 0$.

Then we are justified in making the following substitutions:

$\displaystyle f^\prime \left({c}\right) = \lim_{x \to c} \frac {f \left({x}\right) - f \left({c}\right)} {x - c}$

Part 2
The proof in the other direction is the same.