The Exact Speed of a Falling Stone at One Instant

A speedometer shows speed at one instant, but speed is distance divided by time — and in one instant, both are zero.

The way out is a limit: find the average speed over a tiny time interval $h$, then let $h$ shrink to $0$. The value it settles to is the derivative.

This part covers derivatives from first principles, their meaning as rate and slope, the algebra of derivatives, and derivatives of polynomials, $\sin x$ and $\cos x$.

**The derivative of $f$ is $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, provided this limit exists; finding it directly from this definition is called differentiating from first principles.

Worked example 1 — at a point.** For $f(x)=3x^2$ at $x=2$:

$$f^{\prime }(2)=\underset{h\to 0}{\lim }\frac{3(2+h{)}^2-12}{h}=\underset{h\to 0}{\lim }\frac{12h+3h^2}{h}=\underset{h\to 0}{\lim }(12+3h)=12$$

**Worked example 2 — $f(x)=\frac{1}{x}$.**

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{1}{h}\left(\frac{1}{x+h}-\frac{1}{x}\right)=\underset{h\to 0}{\lim }\frac{-1}{x(x+h)}=-\frac{1}{x^2}$$

**Worked example 3 — $f(x)=\sqrt{x}$, by rationalising.**

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\sqrt{x+h}-\sqrt{x}}{h}=\underset{h\to 0}{\lim }\frac{1}{\sqrt{x+h}+\sqrt{x}}=\frac{1}{2\sqrt{x}}$$

An everyday example. Checking the speed of a train between two stations a few seconds apart, then a fraction of a second apart, is the first-principles method in practice.

The substance. Every standard formula comes from this definition — the rules that follow are shortcuts, not replacements.

**$\frac{f(x+h)-f(x)}{h}$ is an average rate of change and also the slope of a chord; as $h\to 0$ the chord turns into the tangent, so $f^{\prime }(a)$ is both the instantaneous rate at $a$ and the slope of the tangent at $(a,f(a))$.

Worked example 1 — speed.** A stone falls $s=5t^2$ metres in $t$ seconds.

$$\text{average speed from }t=3\text{ to }3.1=\frac{5(3.1{)}^2-5(3{)}^2}{0.1}=\frac{48.05-45}{0.1}=30.5\text{ m/s}$$

$$\frac{ds}{dt}=10t\Rightarrow \text{speed at }t=3\text{ is }30\text{ m/s}$$

Worked example 2 — slope. For $y=x^3$ at $(2,8)$, the slope of the tangent is $3x^2=12$, so the tangent line is

$$y-8=12(x-2)\Rightarrow y=12x-16$$

Worked example 3 — area. For a circle, $A=\pi r^2$ gives $\frac{dA}{dr}=2\pi r$. At $r=5$ cm, the area grows at $10\pi \approx 31.4$ square cm per cm of radius.

An everyday example. Rolling out a roti: each extra millimetre of radius adds more area when the roti is already large, just as $2\pi r$ grows with $r$.

The substance. A positive derivative means the function is rising at that point; a negative one means it is falling.

**For differentiable $u$ and $v$: $(u\pm v{)}^{\prime }=u^{\prime }\pm v^{\prime }$, $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$, and ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$ where $v\ne 0$.

Worked example 1 — product rule, checked by expanding.**

$$\frac{d}{dx}\left[(x^2+1)(x^3-2)\right]=2x(x^3-2)+(x^2+1)(3x^2)=5x^4+3x^2-4x$$

Expanding first gives $x^5+x^3-2x^2-2$, whose derivative is the same.

Worked example 2 — quotient rule.

$$\frac{d}{dx}\left(\frac{x+1}{x-1}\right)=\frac{(1)(x-1)-(x+1)(1)}{(x-1{)}^2}=\frac{-2}{(x-1{)}^2}$$

**Worked example 3 — $\tan x$.**

$$\frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)=\frac{\cos x\cos x-\sin x(-\sin x)}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}={\sec }^{2}x$$

An everyday example. A shop's takings are price times quantity sold. If both change, the change in takings is (change in price) times quantity plus price times (change in quantity) — the product rule.

The substance. **$(uv{)}^{\prime }$ is not $u^{\prime }{v}^{\prime }$** — for $x\cdot x$ that would give $1$ instead of $2x$.

**Using ${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$ gives $\frac{d}{dx}{x}^n=n{x}^{n-1}$, so a polynomial is differentiated term by term; first principles with $\frac{\sin \theta }{\theta }\to 1$ give $\frac{d}{dx}\sin x=\cos x$ and $\frac{d}{dx}\cos x=-\sin x$.

Worked example 1 — polynomial.**

$$\frac{d}{dx}(4x^5-3x^2+7x-9)=20x^4-6x+7$$

**Worked example 2 — $\sin x$ from first principles.**

$$\frac{\sin (x+h)-\sin x}{h}=\frac{2\cos \left(x+\frac{h}{2}\right)\sin \frac{h}{2}}{h}=\cos \left(x+\frac{h}{2}\right)\cdot \frac{\sin (h/2)}{h/2}\to \cos x$$

The same steps with $\cos (x+h)-\cos x=-2\sin \left(x+\frac{h}{2}\right)\sin \frac{h}{2}$ give $-\sin x$.

Worked example 3 — combining.

$$\frac{d}{dx}(3\sin x-2\cos x)=3\cos x+2\sin x\text{value at }x=0\text{ is }3$$

Worked example 4 — product.

$$\frac{d}{dx}(x^2\sin x)=2x\sin x+x^2\cos x$$

An everyday example. On a giant wheel at a mela, your height follows a sine curve; your vertical speed follows a cosine curve — greatest as you pass the middle height and zero at the very top.

The substance. **The minus sign belongs to the derivative of $\cos x$**, not of $\sin x$.