Chapter 2 · Introduction

Chapter 2 — Derivatives

The instantaneous rate of change: slopes of tangent lines, and the rules that make derivatives quick to compute.

Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science and engineering, and it is critical to analysis in business and health as well. In this chapter we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions; applications — tangent lines, motion and other rates of change — follow in Chapter 3.

Slope of a secant→Limit → tangent→Rules (power, product, quotient)→Chain rule→Implicit→Parametric
🎯 By the end of this chapter you can…
  • find a derivative from the limit definition and interpret it as the slope of a tangent line;
  • differentiate polynomial, rational, trigonometric, exponential, logarithmic and inverse-trig functions with the rules;
  • apply the chain rule, implicit and logarithmic differentiation, and differentiate parametric curves.
  • (Tangent- and normal-line equations are collected in 3.1 Tangent and Normal Lines; motion and rates of change in 3.2 Derivatives as a Rate of Change.)
Teaser — From Secant to Tangent

For $f(x)=x^2$ at $a=1$, the secant through $(1,1)$ and $(1+h,(1+h)^2)$ has slope $2+h$. Shrink $h$ and the secant becomes the tangent, with slope $f^\prime(1)=2$.

$h$—
secant slope—
tangent slope—
$f^\prime(a)=\displaystyle\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$ — the definition that the whole chapter is built on. Start 2.1 →

Sections in this chapter

2.1

Defining the Derivative

Secants and the slope of the tangent, the derivative at a point, average vs instantaneous rates, and the derivative as a function.

$f'(a)=\displaystyle\lim_{h\to0}\tfrac{f(a+h)-f(a)}{h}$
Open section →2.2

Differentiation Rules

Constant, power, sum, product and quotient rules; trig, exponential, log and inverse-trig derivatives.

$(fg)^\prime=f^\prime g+fg^\prime$
Open section →2.3

The Chain Rule

Differentiate compositions: outside derivative times inside derivative.

$\tfrac{dy}{dx}=\tfrac{dy}{du}\cdot\tfrac{du}{dx}$
Open section →2.4

Implicit Differentiation

Slopes of curves like circles; second derivatives implicitly; logarithmic differentiation.

$x^2+y^2=25\Rightarrow y^\prime=-\tfrac xy$
Open section →2.5

Parametric Differentiation

Slopes and second derivatives of curves given by $x(t)$, $y(t)$.

$\tfrac{dy}{dx}=\tfrac{dy/dt}{dx/dt}$
Open section →
✅ Before you start
You will need limits (Chapter 1), expanding $(a+h)^n$, exponent laws and trigonometric values (0.1 §7, §9).
← 1.4 Limits Involving Infinity2.1 Defining the Derivative →