Definition of the Derivative

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

Alternate form: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

• Geometric: slope of the tangent line at $x = a$.
• Physical: instantaneous rate of change.
• $f'(a)$ is the instantaneous velocity if $f$ measures position.

• Corner (V-shape).
• Cusp (sharp point).
• Vertical tangent.
• Discontinuity.

Differentiable → Continuous (but not vice versa).

Find $f'(x)$ for $f(x) = x^2$ from the definition.

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