Derivatives

The instantaneous rate of change of a function — defined as the limit of the difference quotient and interpreted as the slope of a tangent line.

Practice exercises →

Derivative as slope of the tangent line — f(x) = x²

a = 1.002

Slope of tangent at x = a equals f′(a) = 2a

Definition

The derivative of a function $f$ at a point $x$ is the instantaneous rate of change of $f$ at that point. Geometrically, it is the slope of the tangent line to the graph of $f$ at $(x, f(x))$ .

It is defined as a limit:

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

This expression is the slope of the secant line through $(x, f(x))$ and $(x+h, f(x+h))$ as $h$ shrinks to zero.

Notation: $f'(x)$ , $\dfrac{df}{dx}$ , $\dfrac{d}{dx}[f(x)]$ , and $\dot{f}$ (in physics) all mean the same thing.

Derivative of x²

Find the derivative of $f(x) = x^2$ .

$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0}(2x + h) = 2x$

So $\frac{d}{dx}[x^2] = 2x$ . At $x = 3$ , the tangent line has slope $6$ .

Slope of the tangent

Find the slope of the tangent to $f(x) = x^3$ at $x = 2$ .

Related concepts

Needs first

Limits

Fundamental Theorem of Calculus Implicit Differentiation Integrals Newton's Method Related Rates Taylor Series

Uses this

Euler's Method Series & Convergence

View in full concept graph →