How Calculus Measures Speed at a Single Instant

A speedometer shows speed at one instant, not over a whole journey. The derivative captures that idea for any changing quantity — the rate at which a function changes at a single point, which is also the slope of its graph there.

This lesson covers the derivative as rate of change and slope, first principles, the sum, product and quotient rules, and the standard derivative formulae.

**The derivative of $f(x)$ at $x=a$ is $f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$, the limit of average rates of change over shorter and shorter intervals, and geometrically it is the slope of the tangent to the curve at $(a,f(a))$.

From average to instantaneous.** The fraction $\frac{f(a+h)-f(a)}{h}$ is the slope of the chord joining two points on the curve. As $h\to 0$, the second point slides towards the first and the chord turns into the tangent.

Worked example. A stone falls $s=4.9t^2$ metres in t seconds. Its average speed from $t=2$ to $t=2+h$ is

$$\frac{4.9(2+h{)}^2-4.9\times 4}{h}=19.6+4.9h$$

For $h=0.1$ this is 20.09 m/s, and as $h\to 0$ it approaches 19.6 m/s — the speed at exactly 2 seconds.

Tangent line. For $y=x^2$ at $(3,9)$ the slope is $2\times 3=6$, so the tangent is $y-9=6(x-3)$, that is $y=6x-9$.

An everyday example. A speed gun reading a fast bowler's delivery at release gives an instantaneous speed, not the average over the whole pitch.

The substance. A derivative can fail to exist at a sharp corner — $|x|$ has no single tangent at $x=0$.

**To differentiate from first principles, write $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, simplify the numerator until h cancels, and then let h approach 0.

Worked example (algebraic).** Differentiate $f(x)=x^3$:

$$\frac{(x+h{)}^3-x^3}{h}=\frac{3x^{2}h+3x{h}^2+h^3}{h}=3x^2+3xh+h^2\to 3x^2$$

Worked example (reciprocal). Differentiate $f(x)=\frac{1}{x}$:

$$\frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\frac{-h}{hx(x+h)}=\frac{-1}{x(x+h)}\to -\frac{1}{x^2}$$

Worked example (trigonometric). Differentiate $f(x)=\sin x$ using the sum-to-product formula:

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

since $\frac{\sin \theta }{\theta }\to 1$ as $\theta \to 0$.

An everyday example. How fast the area of a square floor tile grows as its side grows — for a side of 10 cm, $\frac{d}{dx}(x^2)=2x=20$ square centimetres per centimetre.

The substance. **First principles always begin with a $\frac{0}{0}$ form** — the algebra exists only to cancel that h.

**The derivative of a sum or difference is the sum or difference of the derivatives, the product rule is $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$, and the quotient rule is ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$ wherever $v\ne 0$.

The rules:**

- $(u\pm v{)}^{\prime }=u^{\prime }\pm v^{\prime }$ and $(ku{)}^{\prime }=k{u}^{\prime }$ for a constant k
- $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$
- ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$

Why the product rule works. Writing $u(x+h)v(x+h)-u(x)v(x)=[u(x+h)-u(x)]v(x+h)+u(x)[v(x+h)-v(x)]$, dividing by h and letting $h\to 0$ gives $u^{\prime }v+u{v}^{\prime }$.

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

Worked example (quotient). For $y=\frac{x+1}{x-1}$:

$$\frac{dy}{dx}=\frac{(1)(x-1)-(x+1)(1)}{(x-1{)}^2}=\frac{-2}{(x-1{)}^2}$$

At $x=3$ the slope is $-\frac{2}{4}=-0.5$.

An everyday example. A shop's revenue is price times quantity sold, so when both change over a festival season, the rate of change of revenue follows the product rule.

The substance. The derivative of a product is not the product of the derivatives — $x\times x$ has derivative $2x$, not $1\times 1$.

**The power rule gives $\frac{d}{dx}(x^n)=n{x}^{n-1}$, and the trigonometric derivatives are $\sin x\to \cos x$, $\cos x\to -\sin x$, $\tan x\to {\sec }^{2}x$, $\cot x\to -{\text{cosec}}^{2}x$, $\sec x\to \sec x\tan x$ and $\text{cosec}x\to -\text{cosec}x\cot x$.

Worked example (polynomial).** For $f(x)=4x^3-5x^2+7x-2$:

$$f^{\prime }(x)=12x^2-10x+7,f^{\prime }(2)=48-20+7=35$$

Worked example (trigonometric). By the quotient rule,

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

so at $x=\frac{\pi }{4}$ the slope of $y=\tan x$ is 2.

Worked example (fractional and negative powers). $\frac{d}{dx}\left(\sqrt{x}+\frac{1}{x^2}\right)=\frac{1}{2\sqrt{x}}-\frac{2}{x^3}$; at $x=4$ this is $0.25-0.03125=0.21875$.

An everyday example. **A car whose position is $s=2t^3-3t^2$ metres** has velocity $6t^2-6t$ m/s, so at $t=3$ s it is moving at 36 m/s.

The substance. The power rule works for negative and fractional powers too — rewrite roots and reciprocals as powers first.