Calculus - Differentiation and its applications — Higher derivatives of the function

About this lesson

This unit focuses on higher-order derivatives of functions. It begins by explaining that if y = f(x) is differentiable, its first derivative is a new function, and if that is differentiable, its derivative is the second derivative, denoted y'' or d²y/dx². Repeating this process yields third and higher derivatives, with the nth derivative written as y^(n) or f^(n)(x). The unit emphasizes notation, distinguishing between d²y/dx² (second derivative) and (dy/dx)² (square of the first derivative). It provides examples of finding second and third derivatives for polynomial, rational, trigonometric, and radical functions, as well as for implicit functions and parametric equations. The text includes worked examples, 'try to solve' exercises, and a critical thinking question about patterns in derivatives of y = sin(ax). By the end, students should be able to compute higher-order derivatives for various function types, including implicit and parametric forms, and prove related identities.