Mathematics I · Week 8 — Derivatives and critical points

Week map

Derivative notation

• $f'(x)$ or $\frac{dy}{dx}$ → derivative of $f$ at $x$ → instantaneous rate of change → slope of tangent line at $x$.
• $(x^n)' = nx^{n-1}$ → power rule → bring exponent down, reduce power by one.
• $(cf)' = c \cdot f'$ → constant multiple.
• $(f + g)' = f' + g'$ → sum rule.

Chain and product (when needed)

• Chain: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$ — derivative of outside × derivative of inside.
• Product: $(fg)' = f'g + fg'$.

Tangent line

• Line through $(a, f(a))$ with slope $m = f'(a)$: $y - f(a) = f'(a)(x - a)$.
• Tangent approximates curve locally near $x = a$.

Critical points

• Critical point → where $f'(x) = 0$ or $f'(x)$ undefined.
• First derivative test: sign of $f'$ changes from $+$ to $-$ at critical point → local max; $-$ to $+$ → local min.
• Increasing: $f' > 0$; decreasing: $f' < 0$.

Pattern families

Easy — Power and sum rule

Medium — Tangent line and evaluate derivative

Hard — Critical points and extrema

Worked mini-examples

Traps

• Forgetting to multiply by inner derivative in chain rule.
• Critical point assumed to be max/min without sign test.
• Tangent uses wrong point — must pass through $(a, f(a))$.
• $(fg)' written as$f'g'$ — product rule differs.
• Derivative of constant not zero in long expressions.

Diagnostic (try yourself)

1. Find $f'(x)$ for $f(x) = 4x^3 - 5x^2 + x - 9$.
2. Differentiate $g(x) = \sqrt{x} + 1/x$ (rewrite powers first).
3. Find the equation of the tangent line to $f(x) = x^2$ at $x = -2$.
4. For $f(x) = x^3 - 3x$, find all critical points.
5. If $f'(x) > 0$ on an interval, is $f$ increasing or decreasing there?