Slope of Parabola at Point

Theorem

Let $a$ be a constant.

Let $T$ be the parabola which is the locus of points $\tuple {x, y}$ satisfying $y = a x^2$.

The slope of the tangent to $y = a x^2$ at $x = c$ is $2 a c$.

Proof

By Derivative of Power of Function the derivative of $y = ax^2$ is:

$\dfrac {\d y} {\d x} = 2 a x$

So the derivative of $y = a x^2$ at $x = c$ is $2 a c$.

The result follows from Slope of Curve at Point equals Derivative.

$\blacksquare$