Slope (or Gradient) of Tangent and Normal

Take a curve $y = f(x)$, a point $A(x, f(x))$ on it, and a nearby point $B(x + \delta x,\, f(x + \delta x))$. The straight line joining $A$ and $B$ is a secant, and its slope is $\dfrac{f(x+\delta x) - f(x)}{\delta x} = \tan\theta$, where $\theta$ is the secant's angle of inclination.

Now let $B$ slide along the curve towards $A$, so that $\delta x \to 0$. If the secant $AB$ settles into a limiting position as $B$ approaches $A$, that limiting line is the tangent to the curve at $A$ — and, because the derivative is defined as exactly this limit, the slope of the (non-vertical) tangent at a point $A(x_0, y_0)$ is

The normal to the curve at a point is the line through that point perpendicular to the tangent there. Since perpendicular lines have slopes that are negative reciprocals of one another, the slope of the normal follows immediately from the slope of the tangent: …