Introduction

Differential calculus was originally called infinitesimal calculus, because its core idea is to break something into infinitesimally small parts to see how it changes. Its applications to physics and astronomy are as old as the origin of modern science itself; through the 18th century these applications multiplied, and by its end Laplace and Lagrange had brought the whole study of forces under the umbrella of mathematical analysis. Dirichlet, Riemann, von Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami and many other leading physicists of the era carried the applications of differentiation further still.

Differential calculus today reaches well beyond geometry and dynamics: derivatives of a function representing cost, material strength, profit or similar quantities let us determine where that function is increasing or decreasing (its monotonicity), and from there its extreme values; derivatives show up throughout engineering and science modelling problems, and in the social and medical sciences too.

What this chapter builds, using only the first two derivatives of $f(x)$:

• the nature of the function — its monotonicity, convexity and concavity;
• sketching the curve $y=f(x)$;
• the local extrema (maxima or minima) of $f(x)$;

and, using still higher derivatives where they exist, the series expansion of $f(x)$ about a point.

Learning objectives. By the end of this chapter you should be able to: apply derivatives to geometrical problems (tangents, normals, angle between curves); use derivatives on physical problems (rates of change, related rates); identify the nature of curves — monotonicity, convexity and concavity; model real-world problems to compute extreme values using derivatives; and trace curves of polynomials and other functions.