The derivative

The manager wants a big screen at the bottom of the Big Drop showing each rider’s top speed, like a radar gun. Lesson 2 found the speed at $t = 1$: the average speed from 1 to $1 + k$ approaches 9.8 m/s as $k \to 0$.

That number has a name: the derivative of $h$ at $t = 1$, written $h'(1)$. In 1665–66, a young Isaac Newton developed the idea (he called it a “fluxion”) to describe moving planets; around 1675, Gottfried Leibniz invented it independently and gave it the $\frac{dy}{dx}$ notation we still use. Their two methods became calculus, and the derivative is its first great idea.

Today you’ll see the derivative two ways, as a speed and as a slope, and then write the speed display.

The real words

• The Derivative of $f$ at $a$:

  $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

  if that limit exists. It’s the instantaneous rate of change, measured in (units of $f$) per (units of $x$).

• Leibniz notation: $\frac{df}{dx}$ or $\frac{dy}{dx}$, read “dy by dx”: a tiny change in $y$ over a tiny change in $x$. For time, physicists often write $\dot{h}$.

• The Tangent line at $a$ passes through $(a, f(a))$ with slope $f'(a)$:

  $$y = f(a) + f'(a)\,(x - a)$$

• Working out the Big Drop’s derivative from the definition:

  $$\begin{aligned} h(t + k) - h(t) &= -4.9\big((t + k)^2 - t^2\big) \\ &= -4.9(2tk + k^2) \end{aligned}$$

  Divide by $k$: $-9.8t - 4.9k$. As $k \to 0$ this approaches $-9.8t$. So $h'(t) = -9.8t$ for every $t$.

• The derivative function $f'(x)$ gives the slope at every $x$. Where $f$ rises, $f' > 0$; where it falls, $f' < 0$; at a flat peak or valley, $f' = 0$.

• Not every function has a derivative everywhere: $|x|$ has a sharp corner at 0, where the slopes from the left (−1) and right (1) disagree.