Slope of Curve at Point equals Derivative
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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 $P$.
$\blacksquare$
Sources
- 1956: E.L. Ince: Integration of Ordinary Differential Equations (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): derivative
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): derivative