The Timpson Lyceum · Calculus 1 (2024)

Differentiation Rules, Rates of Change, & Derivatives of Trig Functions - Lecture 9 (Calculus 1)

Overview

This lecture trades the limit definition of the derivative for a toolkit of fast differentiation rules and proves each one rigorously. You learn the constant, power, constant-multiple, and sum rules, then the product and quotient rules, and how to take higher-order derivatives. The second half reinterprets the derivative as a rate of change, defining position, velocity, speed, acceleration, and jerk, and applies these to free-fall problems. Finally, the basic derivatives of the trigonometric functions $\sin x$ , $\cos x$ , and $\tan x$ are derived and applied. By the end you should be able to differentiate most polynomial, rational, and trigonometric expressions quickly without resorting to limits.

Key Concepts

Definition of the derivative (recap). All of these notations mean the same thing — the instantaneous rate of change (slope) of $y = f(x)$ :

$f'(x) = y' = \dfrac{dy}{dx} = \displaystyle\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$
Equivalent form: $f'(x) = \displaystyle\lim_{z \to x}\frac{f(x)-f(z)}{x-z}$
Derivative at a point $c$ : $f'(c) = \displaystyle\lim_{x \to c}\frac{f(x)-f(c)}{x-c}$

The basic rules. For $u = f(x)$ , $v = g(x)$ , and constant $c$ :

Constant rule: $\dfrac{d}{dx}(c) = 0$ — a constant function has zero slope.
Power rule: $\dfrac{d}{dx}\,x^{n} = n\,x^{\,n-1}$ — bring the exponent down, then subtract one. It holds for any real $n$ (proved here only for natural numbers $n$ ). Example: $\frac{d}{dx}x^5 = 5x^4$ .
Constant-multiple rule: $\dfrac{d}{dx}\big(c\,u\big) = c\,\dfrac{du}{dx}$ — pull constants outside.
Sum rule: $\dfrac{d}{dx}(u+v) = \dfrac{du}{dx} + \dfrac{dv}{dx}$ — differentiate term by term.

Product rule. The derivative of a product is not the product of the derivatives: $\frac{d}{dx}(uv) = u\,\frac{dv}{dx} + v\,\frac{du}{dx}$ Mnemonic: the first times the derivative of the second, plus the second times the derivative of the first. The proof adds and subtracts $f(x+h)\,g(x)$ in the numerator, then factors.

Quotient rule. $\frac{d}{dx}\!\left(\frac{u}{v}\right) = \frac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^{2}}$ Mnemonic: low D-high minus high D-low, square the bottom and away we go. Order matters here (unlike the product rule) because of the minus sign.

Higher-order derivatives. You can differentiate repeatedly. Notations: $f''(x) = \frac{d^2y}{dx^2}$ for the second derivative; in general $f^{(n)}(x) = \frac{d^n y}{dx^n}$ (the parentheses on $f^{(n)}$ distinguish it from a power). For a polynomial of degree $k$ , the $(k{+}1)$ -th derivative is $0$ .

Rates of change. A derivative measures how one quantity changes with respect to another. For a circle's area $A = \pi r^2$ , the rate of change of area with respect to radius is $\frac{dA}{dr} = 2\pi r$ (units: m²/m = m).

Motion definitions. Let $s(t)$ be position at time $t$ :

Displacement over $[t,\,t+\Delta t]$ : $s(t+\Delta t) - s(t)$ .
Velocity (instantaneous): $v(t) = \dfrac{ds}{dt} = \displaystyle\lim_{\Delta t \to 0}\frac{s(t+\Delta t)-s(t)}{\Delta t}$ . Has direction (can be negative).
Speed: $\left|\dfrac{ds}{dt}\right|$ — the absolute value of velocity; never negative.
Acceleration: $a(t) = \dfrac{dv}{dt} = \dfrac{d^2 s}{dt^2}$ .
Jerk: $\dfrac{da}{dt} = \dfrac{d^3 s}{dt^3}$ .

Free fall. Near Earth's surface, acceleration due to gravity is constant: $g = 9.8\ \text{m/s}^2$ (or $32\ \text{ft/s}^2$ ). An object's fall distance is $s(t) = \tfrac{1}{2}g\,t^2$ (the $\tfrac12$ comes from integrating the constant acceleration twice).

Derivatives of trig functions.

$\dfrac{d}{dx}\sin x = \cos x$
$\dfrac{d}{dx}\cos x = -\sin x$
$\dfrac{d}{dx}\tan x = \sec^2 x$
$\dfrac{d}{dx}\sec x = \sec x \tan x$

The sin/cos derivatives follow from the angle-addition formulas plus the limits $\lim_{h\to 0}\frac{\sin h}{h} = 1$ and $\lim_{h\to 0}\frac{\cos h - 1}{h} = 0$ . The tan and sec derivatives then follow from the quotient rule (using $\tan x = \frac{\sin x}{\cos x}$ , $\sec x = \frac{1}{\cos x}$ , and $\sin^2 x + \cos^2 x = 1$ ).

Worked Example

Quotient rule. Differentiate $y = \dfrac{t^2 - 1}{t^3 + 1}$ .

Rule: $\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) = \dfrac{v\,u' - u\,v'}{v^2}$ , here with $u = t^2-1$ , $v = t^3+1$ .

\begin{aligned} y' &= \frac{(t^3+1)\,\frac{d}{dt}(t^2-1) - (t^2-1)\,\frac{d}{dt}(t^3+1)}{(t^3+1)^2} \\ &= \frac{(t^3+1)(2t) - (t^2-1)(3t^2)}{(t^3+1)^2} \\ &= \frac{2t^4 + 2t - 3t^4 + 3t^2}{(t^3+1)^2} \\ &= \frac{-t^4 + 3t^2 + 2t}{(t^3+1)^2} \end{aligned}

Free-fall rates of change. A rock is launched straight up at $160\ \text{ft/s}$ , so $s(t) = 160t - 16t^2$ (up is positive). Differentiating gives velocity and acceleration, and setting $v = 0$ locates the maximum height:

\begin{aligned} v(t) &= \frac{ds}{dt} = 160 - 32t, \qquad a(t) = \frac{dv}{dt} = -32\ \text{ft/s}^2 \\ 0 &= 160 - 32t \;\Longrightarrow\; t = 5\ \text{s} \quad(\text{apex}) \\ s(5) &= 160(5) - 16(5)^2 = 800 - 400 = 400\ \text{ft (max height)} \end{aligned}

Summary

Differentiation rules replace the limit definition: memorize the constant, power ( $\frac{d}{dx}x^n = n x^{n-1}$ ), constant-multiple, and sum rules for fast differentiation of polynomials.
The product rule is $u v' + v u'$ and the quotient rule is $\frac{v u' - u v'}{v^2}$ — neither is simply the product/quotient of the derivatives.
Derivatives can be taken repeatedly; $f^{(n)}$ denotes the $n$ -th derivative, and differentiating a degree- $k$ polynomial $k+1$ times gives $0$ .
A derivative is a rate of change: in motion, velocity $= s'$ , acceleration $= s''$ , jerk $= s'''$ ; speed is $|v|$ and ignores direction.
Free-fall position is $s(t) = \tfrac12 g t^2$ with $g = 9.8\ \text{m/s}^2$ ( $32\ \text{ft/s}^2$ ); the apex of a thrown object occurs where $v = 0$ .
Core trig derivatives: $\sin x \to \cos x$ , $\cos x \to -\sin x$ , $\tan x \to \sec^2 x$ , $\sec x \to \sec x\tan x$ .