Derivatives as local rate of change

Tangent Lines and Meaning

The derivative of $f$ at $x$ gives the instantaneous rate of change and is defined as 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 at $(a, f(a))$. The tangent line equation is:

$$y - f(a) = f'(a)(x - a)$$

This formula acts as the best linear approximation of $f$ near $a$.

Physical meaning: If $s(t)$ is position, its first derivative $s'(t)$ is velocity and its second derivative $s''(t)$ is acceleration.

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

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 $f'(x)$ causes common errors.

Rules and Differentiability

Quickly compute common derivatives using these foundational rules:

Differentiability vs continuity compares whether smooth rates exist versus unbroken paths:

A function must be continuous to be differentiable, but a sharp corner breaks differentiability while remaining continuous.