Week 8 — derivatives & critical points

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Week map

Classify → Represent → Execute → Trap-check

• Recognize: Ask: What does $f'(2)$ represent geometrically?
• Procedure: Differentiate term by term. Constants vanish; bring down exponent for powers. Rewrite roots and reciprocals as powers before differentiating.
• Variations / traps: Watch for: Forgetting chain rule on $(3x+1)^5$.

Formula chain (compressed)

f′(x)=limit of difference quotient → power/rule derivatives → f′=0 critical → max/min.

1. Definition — f′(x)=lim_{h→0}(f(x+h)−f(x))/h — from first principles
2. Power rule — d/dx x^n = n x^{n−1} — polynomial terms
3. Tangent slope — m_tan = f′(a) — slope at x=a
4. Critical point — f′(x)=0 or undefined — candidate extrema
5. L'Hôpital — lim f/g = lim f′/g′ — 0/0 or ∞/∞ indeterminate

Deep study

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?

ChatGPT prep archive

• Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

• Derivative $f'(x)$: instantaneous rate of change; slope of tangent at $x$.
• Rules: power, sum, product, quotient, chain for compositions.
• Tangent line: $y - f(a) = f'(a)(x-a)$.
• Critical points: $f'(x)=0$ or undefined; test max/min with sign chart or second derivative.

Notation & vocabulary

Pattern families

Easy — Power and sum rule

Medium — Tangent line problem

Hard — Critical points and shape

Drill these on the pattern atlas — filter to week 8.

Traps

• Forgetting chain rule on $(3x+1)^5$.
• Critical point where derivative undefined (corner, vertical tangent).
• Confusing $f'(x)=0$ with always a max.
• Quotient rule order in denominator squared term.

Retrieval prompts

1. What does $f'(2)$ represent geometrically?
2. How do you find critical numbers of $f$?
3. When is L'Hôpital's rule relevant at a limit?

Practice loop

1. Read Deep study (if present) or core concepts once.
2. Recite the formula chain without looking.
3. Open one easy pattern on the interactive atlas for week 8.
4. Attempt without solutions; mark studied after an honest try.
5. Say one trap aloud before closing the tab.

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk