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Mathematics I · Week 8 — Derivatives & critical points

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Mathematics I · Week 8 — Derivatives & critical points

1150 words

6 min read

2026-08-16

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Now · Week map

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.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Derivative as rate → rules → tangent line → critical points → optimization sketch

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.

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

Open interactive formula desk · Week 8 tab.

Deep study

Mathematics I · Week 8 — Derivatives and critical points

Deep study for Quiz 2 week 8. Derivative measures instantaneous rate of change; critical points locate where slope is zero or undefined.

Week map

Derivative as slope → differentiation rules → tangent line → critical points → increasing/decreasing → local extrema sketch.

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.

Mini-example: if

f(x) = x^3 - 2x

, then

f'(x) = 3x^2 - 2

. At

x = 1

f'(1) = 1

— tangent slope is 1.

Chain and product (when needed)

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

Mini-example:

(3x^2 + 1)^4

— outer power 4, inner

3x^2+1

. Derivative

= 4(3x^2+1)^3 \cdot 6x

Tangent line

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

Mini-example:

f(x) = x^2

a = 3

f(3) = 9

f'(x) = 2x

f'(3) = 6

. Tangent:

y - 9 = 6(x - 3)

→

y = 6x - 9

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: $f' < 0$ .

Mini-example:

f(x) = x^2 - 4x + 3

f'(x) = 2x - 4 = 0

→

x = 2

f'

negative before 2, positive after → local min at

x = 2

Pattern families

Easy — Power and sum rule

Differentiate term by term. Constants vanish. Rewrite roots and reciprocals as powers first:

\sqrt{x} = x^{1/2}

1/x = x^{-1}

Medium — Tangent line and evaluate derivative

Find

f'(a)

, write tangent equation. Interpret

f'(a)

as rate (velocity, marginal cost) in word problems.

Hard — Critical points and extrema

Solve

f'(x) = 0

. Build sign chart. Classify max/min. Sketch behavior without full graph.

Worked mini-examples

Example 1 — Power rule.

\frac{d}{dx}(5x^4 - 3x + 7) = 20x^3 - 3

Example 2 — Rewrite then differentiate.

f(x) = \frac{1}{x^2} = x^{-2}

→

f'(x) = -2x^{-3} = -2/x^3

Example 3 — Tangent.

f(x) = x^3

a = 1

f(1) = 1

f'(1) = 3

. Tangent:

y = 3x - 2

Example 4 — Critical point.

f(x) = -x^2 + 6x

f'(x) = -2x + 6 = 0

→

x = 3

. Max (parabola opens down).

Example 5 — Chain rule.

f(x) = (2x + 1)^3

f'(x) = 3(2x+1)^2 \cdot 2 = 6(2x+1)^2

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)

Find $f'(x)$ for $f(x) = 4x^3 - 5x^2 + x - 9$ .
Differentiate $g(x) = \sqrt{x} + 1/x$ (rewrite powers first).
Find the equation of the tangent line to $f(x) = x^2$ at $x = -2$ .
For $f(x) = x^3 - 3x$ , find all critical points.
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

Rule	Formula
Power	$(x^n)' = nx^{n-1}$
Chain	$(f(g(x)))' = f'(g(x))g'(x)$
Tangent	slope $m = f'(a)$ at $x=a$

Pattern families

Easy — Power and sum rule

Differentiate term by term. Constants vanish; bring down exponent for powers. Rewrite roots and reciprocals as powers before differentiating.

Medium — Tangent line problem

Find

f'(a)

and

f(a)

. Plug into point-slope form. Simplify to slope-intercept if needed. Domain of tangent matches domain of derivative.

Hard — Critical points and shape

Solve

f'(x)=0

on domain. Classify with sign chart of

f'

f''

. State intervals of increase/decrease. For Quiz 2 scope, justify max/min in words from derivative sign change.

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

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

Practice loop

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