Slope of Curve at Point equals Derivative

Theorem

Let $\CC$ be a curve embedded in the Cartesian plane described using the equation:

$y = \map f x$

where $f$ is a real function.

Let there exist a unique tangent $\TT$ to $\CC$ at a point $P = \tuple {x_0, y_0}$ on $\CC$.

Then the slope of $\CC$ at $P$ is equal to the derivative of $f$ at $P$.

Proof

We have been given that there exists a unique tangent $\TT$ to $\CC$ at $P$.

By definition of tangent, $\TT$ has a slope $M$ given by:

$m = \ds \lim_{h \mathop \to 0} \frac {\map f {x_0 + h} - \map f {x_0} } h$

This is the definition of the derivative of $f$ at $P$.

$\blacksquare$