Derivatives: change right now

The idea

A derivative is the slope you see when the change in $x$ shrinks to an instant. Average change uses two points on a curve; as those points move together, the secant line connecting them approaches the tangent line at a single point. In optimization, the sign of the derivative tells you which direction is uphill.

For nested functions, use the chain rule:

$\frac{d}{dx}f(g(x))=f'(g(x))g'(x).$

Differentiate the outside while leaving the inside in place, then multiply by the derivative of the inside. For example:

\frac{d}{dx}\sin(x^2) = \cos(x^2)\cdot 2x.

Real-world: reading a car's speedometer

Average speed uses a whole time interval, but a speedometer estimates the car's speed right now — the instantaneous rate at which position is changing. If position is $s(t) = 5t^2$ metres, then $s'(t) = 10t$ , so at $t=3$ seconds the car is going $s'(3) = 30$ m/s: the derivative converts position into instantaneous velocity.

At x=2, the slope of x² is 2x = 4.

Worked example

Let $f(x) = x^2$
At $x=3$ , use a nearby point $x = 3+h$
The change ratio is $\dfrac{(3+h)^2 - 9}{h} = 6+h$
As $h \to 0$ , the ratio $\to 6$ ; therefore $f'(3) = 6$

Cheat sheet

$f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$

The derivative is the slope of the tangent line at a single point.
$f'(x) > 0$ means increasing; $f'(x) < 0$ means decreasing; $f'(x) = 0$ flags a candidate max or min.
For $f(x) = x^2$ , $f'(x) = 2x$ — the general pattern is $\dfrac{d}{dx} x^n = nx^{n-1}$ .
Chain rule: $\dfrac{d}{dx}f(g(x))=f'(g(x))g'(x)$ for nested functions.
Velocity is the derivative of position; acceleration is the derivative of velocity.