Applications of Differentiation

The derivative tells us the gradient of a curve, and the gradient encodes a wealth of geometric and physical information: which way a curve is heading, where it turns, and how to optimize a quantity. This lesson puts the derivative to work.

Tangents and normals. At a point $(x_1, y_1)$ on the curve $y = f(x)$, the gradient of the tangent is $m = f'(x_1)$. The tangent line is $$y - y_1 = m(x - x_1).$$ The normal is perpendicular to the tangent, so its gradient is $-\frac{1}{m}$ (provided $m \neq 0$), and its equation is $y - y_1 = -\frac{1}{m}(x - x_1)$.

Worked example (tangent and normal). Find the tangent and normal to $y = x^2 - 3x$ at $x = 4$. First $y(4) = 16 - 12 = 4$, so the point is $(4, 4)$. Then $\frac{dy}{dx} = 2x - 3$, so at $x = 4$ the gradient is $m = 5$. Tangent: $y - 4 = 5(x