Unit 2: Differentiation: Definition and Basic Derivative Rules

Executive Summary

The derivative is the instantaneous rate of change — the slope of the tangent line at a point. Unit 2 defines it and builds the basic rules for computing it.

The definition

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. The average rate of change over $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$; the instantaneous rate is the limit as the interval shrinks to a point.

Differentiability

Differentiable at a point means the derivative exists there — a unique tangent slope. Differentiability implies continuity, but continuity does NOT imply differentiability (corners, cusps, and vertical tangents break it).

Basic rules

Power rule: $\frac{d}{dx} x^n = nx^{n-1}$. Constants, sums, differences, and constant multiples differentiate term by term. Memorize $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$, $\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(\ln x) = \frac{1}{x}$.

Product and quotient rules

Product: $(fg)' = f'g + fg'$. Quotient: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ — numerator order matters, it is "low d-high minus high d-low."

Exam traps

$f'(a)$ is a number (slope); $f'(x)$ is a function. The product rule is NOT $f'g'$. Never confuse $\frac{d}{dx}x^n$ with $\frac{d}{dx}a^x$. Differentiability requires both sides of the limit to agree — a corner has two different one-sided slopes.