Introduction

In the previous chapter we studied how to differentiate composite functions, inverse trigonometric functions, logarithmic functions and parametric functions, and we learned that the derivative $dy/dx$ at a point is precisely the slope of the tangent to the curve $y=f(x)$ at that point.

This chapter builds on that single geometric fact to develop six distinct applications of differentiation:

1. Geometry — using the derivative to write down the equations of the tangent and normal to a curve at a given point.
2. Rate measure — using the derivative to describe how fast one changing quantity varies with respect to another (or with respect to time), which underlies problems about expanding balloons, sliding ladders, moving shadows, and similar situations.
3. Approximations — using the tangent line as a stand-in for the curve very close to a point, to estimate the value of a function (a square root, a trigonometric value, a logarithm, etc.) that is otherwise hard to compute exactly.
4. Rolle's Theorem and Lagrange's Mean Value Theorem — two foundational existence results about differentiable functions, guaranteeing a point where the tangent is horizontal (Rolle's) or parallel to a given chord (Lagrange's).
5. Increasing and decreasing functions — using the sign of the derivative to determine on which intervals a function is rising or falling.
6. Maxima and minima — using the derivative (and its sign changes, or the second derivative) to locate the highest and lowest points of a function's graph in a neighbourhood, and to solve practical optimisation problems (largest area, least surface, and so on).