Derivative of Curve at Point

Theorem
Let $f: \R \to \R$ be a real function.

Let the graph $G$ of $f$ be depicted on a Cartesian plane.

Then the derivative of $f$ at $x = \xi$ is equal to the tangent to $G$ at $x = \xi$.

Proof
Let $f: \R \to \R$ be a real function.


 * DerivativeOfCurve.png

Let the graph $G$ of $f$ be depicted on a Cartesian plane.

Let $A = \left({\xi, f \left({\xi}\right)}\right)$ be a point on $G$.

Consider the secant $AB$ to $G$ where $B = \left({\xi + h, f \left({\xi + h}\right)}\right)$.

From Slope of Secant, the slope of $AB$ is given by:
 * $\dfrac {f \left({x + h}\right) - f \left({x}\right)} h$

By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$.