The Derivative

What is a Derivative?

The derivative measures how fast a function changes at an instant. It is the limit of average rates of change as intervals shrink:

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

The fraction is the slope of the secant line. As $h \to 0$, it becomes the tangent line.

Notation includes $f'(x)$, $\frac{df}{dx}$, $\frac{dy}{dx}$, and $Df$. A function is differentiable if this limit exists.

Pitfall: Do not confuse the derivative (slope) with the function value (height). $f'(x) = 0$ means the graph is flat, not zero.

Behavior & Differentiability

The sign of the derivative tells you the function's local behavior:

Differentiability is stronger than continuity. Every differentiable function is continuous, but not conversely.

A sharp corner (like $|x|$ at 0) is continuous yet not differentiable because left and right slopes disagree. Vertical tangents and discontinuities also block differentiability.