The Definition of the Derivative

A derivative is simply a function that tells you how steep another function is at every point, or, put another way, how fast it's changing. (The name comes from the fact that it's derived from the original function.) It's the single most useful tool in this course: velocity, marginal cost and growth rates are all derivatives. The formula $\lim_{h\to0}\frac{f(x+h) - f(x)}{h}$ looks dense, but it's just rise over run again: $h$ (a small step to the right), $f(x+h) - f(x)$ (how much the height changes over that step) and $\lim_{h\to0}$ (shrink the step to nothing). It's the same slope-between-two-points idea from the Limits unit, so bear with the symbols.

Back in the first lesson of the Limits unit, we found the slope of $x^2$ at one point by sliding a second point in. That worked, but we had to redo it for every point. The derivative does it for all points at once. Call the second point $x + h$ instead of something fixed, and let $h$ shrink. What comes out isn't a number anymore. It's a formula in $x$ that gives the slope wherever you want it.

To compute it from the definition:

1. Write out $f(x + h)$ and subtract $f(x)$.
2. Simplify until a factor of $h$ cancels with the $h$ in the denominator. (It's the same $\frac00$ cleanup from Computing Limits.)
3. Let $h \to 0$.

Notation. $f'(x)$, $\dfrac{dy}{dx}$, $\dfrac{df}{dx}$ and $\dfrac{d}{dx}f(x)$ all mean the same thing. The $\frac{d}{dx}$ form is a reminder that it's a ratio of tiny changes: a tiny change in $y$ over a tiny change in $x$.

Differentiable means continuous, but not the other way around. Think about why the first part is true: if there's a well-defined slope at a point, the graph can't have a hole or a jump there. The reverse fails because a graph can be all in one piece and still not have one clear slope. A function can be continuous yet have no derivative at:

• a corner, like $|x|$ at 0 (the slopes $-1$ and $1$ from each side disagree),
• a cusp, like $x^{2/3}$ at 0,
• a vertical tangent, like $\sqrt[3]{x}$ at 0, where the slope would have to be infinite.

Numerical code estimates derivatives with this exact quotient and a small $h$: physics engines for velocities, machine learning for checking gradients.