Derivative
Calculus & Advanced MathThe derivative measures the instantaneous rate of change of a function, telling you how steeply the graph rises or falls at any point.
Formula
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Definition
The derivative tells you how fast something is changing at one exact moment. On a graph, it equals the steepness (slope) of the curve at a single point.
Example
If your position is described by $x(t) = t^2$ miles after $t$ hours, then your speed (derivative) at $t = 3$ hours is $6$ mph. The graph is steeper there.
Key Insight
Every time you read a speedometer, you are reading a derivative: the instantaneous rate at which distance is changing.
Definition
The derivative of $f$ at $x$ is the limit of the difference quotient: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, provided the limit exists. Geometrically it is the slope of the tangent line to the curve at that point.
Example
$f(x) = x^3$: $f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} = 3x^2$.
Key Insight
The derivative converts a static formula into a dynamic rate. If $f$ gives position, $f'$ gives velocity, and $f''$ gives acceleration.
Definition
$f$ is differentiable at $a$ if $\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ exists. Differentiability implies continuity, but not vice versa. In higher dimensions, the derivative generalizes to the Jacobian matrix of partial derivatives.
Example
$f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$: the left derivative is $-1$ and the right derivative is $+1$.
Key Insight
Differentiability is a strictly stronger condition than continuity. The Weierstrass function is continuous everywhere but differentiable nowhere, a result that shocked 19th-century mathematicians.