Derivatives as local rate of change

The derivative of $f$ at $x$ is the limit of the difference quotient:

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

Geometric meaning — $f'(a)$ is the slope of the tangent line to the curve at $(a,\,f(a))$.

Physical meaning — If $s(t)$ is position, then $s'(t)$ is velocity and $s''(t)$ is acceleration.

Tangent-line equation

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

This is also the best linear approximation of $f$ near $a$.

Differentiability vs continuity

• Differentiable $\Rightarrow$ Continuous.
• Continuous $\not\Rightarrow$ Differentiable (e.g., $|x|$ at $x=0$).

Quick derivatives

Tip: The sign of $f'(x)$ tells you whether $f$ is increasing ($f'>0$) or decreasing ($f'<0$).

Common pitfall: The derivative at a point is a number (a slope), while the derivative of a function is a new function. Confusing $f'(3)$ with $f'(x)$ muddles most early calculus errors.

Derivatives as Rate of Change

The derivative $f'(x)$ measures the instantaneous rate of change of $f$ at $x$. Geometrically, it is the slope of the tangent line:

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

Where $f'(x) > 0$ the function is increasing; where $f'(x) < 0$ it is decreasing; and $f'(x) = 0$ marks critical points.

The tangent line is the best linear approximation to $f$ near the point of tangency.