Tangents & Normals

Tangent slope f'(a); normal slope -1/f'(a).

The idea

The derivative $f'(a)$ gives the slope of the curve $y = f(x)$ at $x = a$; the tangent and the normal are the two lines built from that slope at the point $(a, f(a))$.

The tangent line at $x = a$ is the line through $(a, f(a))$ whose slope is $f'(a)$. In point–slope form, $y = f(a) + f'(a)(x - a).$ It is the one line that passes through the point in the same direction as the curve, so near $a$ it is the best straight-line approximation to the curve.

The normal line at the same point is the line through $(a, f(a))$ perpendicular to the tangent. Perpendicular slopes multiply to $-1$, so the normal's slope is the negative reciprocal $-\frac{1}{f'(a)},$ and its equation is the same point–slope form built on this slope.

One case is special: where $f'(a) = 0$ the tangent is horizontal, so the normal is vertical. A vertical line has no slope, and we write it as $x = a$.

Ways to work on it

• Walkthrough. Build the tangent line and the perpendicular normal.
• Practice. Compute the slope of the tangent to a parabola at a point.
• Hardest. Tangent to a cubic and its intercept and normal.

Not sure where to start? Take the ten-question placement test.