Introduction

Introduction

The idea of the derivative goes back to the 17th century, when Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed the tools of calculus to describe how quantities change. At its heart, a derivative captures rate of change — for a function $y = f(x)$, a small change $\delta x$ in $x$ produces a corresponding small change $\delta y$ in $y$, and the derivative asks what happens to the ratio $\dfrac{\delta y}{\delta x}$ as $\delta x$ shrinks toward zero.

You already know the derivatives of standard functions such as $\sin x$, $\cos x$, $e^x$, and $\log x$, along with the basic rules of differentiation from the previous standard:

• Sum/Difference rule: if $y = u \pm v$, then $\dfrac{dy}{dx} = \dfrac{du}{dx} \pm \dfrac{dv}{dx}$
• Product rule: if $y = uv$, then $\dfrac{dy}{dx} = u\dfrac{dv}{dx} + v\dfrac{du}{dx}$
• Quotient rule: if $y = \dfrac{u}{v}$ (with $v \ne 0$), then $\dfrac{dy}{dx} = \dfrac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

This chapter builds on that foundation. It extends differentiation to composite functions (a function of a function, like $\sin\sqrt{x}$ or $e^{\tan x}$), inverse functions, logarithmic differentiation, implicit functions, and parametric functions, and also introduces the geometrical meaning of the derivative and higher-order derivatives. A few new differentiation rules are added along the way to handle these cases.