Tangents and normals

The idea

The derivative at a point is the gradient of the tangent there. The normal is perpendicular to the tangent, so its gradient is $-\frac{1}{m}$. A function is increasing where $f'(x) > 0$ and decreasing where $f'(x) < 0$.

1. Find $y$ at the point, if not given.
2. Substitute $x$ into $\frac{dy}{dx}$ for the tangent gradient $m$.
3. Use $y - y_1 = m(x - x_1)$, with $-\frac{1}{m}$ for a normal.

Watch out A gradient is a number at a point; the tangent is a whole line through that point.