Maths I — Week 4: Derivatives and Rules

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Maths I — Week 4: Derivatives and Rules

The Big Idea: If Week 3 was about the shape of curves, Week 4 is about their slope at every point. The derivative f′(x)f'(x) is the instantaneous rate of change — the slope of the tangent line. Every physics equation describing velocity, every economics model describing marginal cost, every ML gradient update uses this single idea.

1. The Derivative from First Principles

The derivative answers: "How fast is this function changing right now?"
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Intuition: Zoom into any smooth curve far enough and it looks like a straight line. That line's slope is the derivative.
If the limit doesn't exist (e.g., at a sharp corner like
∣x∣|x| at x=0x=0), the function is not differentiable at that point — even if it is continuous.

1.1 Differentiability vs. Continuity

PropertyDifferentiableContinuous
Implies the other?Yes → Diff ⟹ ContNo → Cont ⟹ Diff fails at corners
Fails atSharp turns, cusps, vertical tangentsJumps, holes, infinite values

2. Core Differentiation Rules

These are the workhorses. Know them cold.
RuleFormula
Constantddx(c)=0\frac{d}{dx}(c) = 0
Power Ruleddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}
Sum Rule(f±g)′=f′±g′(f \pm g)' = f' \pm g'
Product Rule(uv)′=u′v+uv′(uv)' = u'v + uv'
Quotient Rule(uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
Exponentialddx(ex)=ex\frac{d}{dx}(e^x) = e^x; ddx(ax)=axln⁡a\frac{d}{dx}(a^x) = a^x \ln a
Natural Logddx(ln⁡x)=1x\frac{d}{dx}(\ln x) = \frac{1}{x}
Trigddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x; ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x
The
Quotient Rule denominator is always v2v^2, never vv. The numerator order is u′v−uv′u'v - uv' (top prime first).

3. The Chain Rule

Used when you have a function inside a function (a composite).
ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
Read as: "Derivative of the outer (keeping inner untouched) × Derivative of the inner."

Pattern: Chain Rule Stack

Question: Differentiate y=sin⁡(x2+5)y = \sin(x^2 + 5).

Abstract Solution (Strategy)

  1. [Identify layers]: Outer = sin⁡(⋅)\sin(\cdot), Inner = x2+5x^2 + 5.
  2. [Outer derivative]: cos⁡(⋅)\cos(\cdot) (keeping inner intact).
  3. [Inner derivative]: 2x2x.

Procedure

  • Step 1 — Outer: cos⁡(x2+5)\cos(x^2 + 5).
  • Step 2 — Inner: ddx(x2+5)=2x\frac{d}{dx}(x^2 + 5) = 2x.
  • Step 3 — Multiply: f′(x)=2xcos⁡(x2+5)f'(x) = 2x \cos(x^2 + 5).

4. Implicit Differentiation

When you can't isolate yy (e.g., x2+y2=25x^2 + y^2 = 25), differentiate both sides with respect to xx. Every yy term gets a dydx\frac{dy}{dx} (chain rule applied to yy).

Procedure Template

  1. Differentiate both sides with respect to xx.
  2. When differentiating a yy-term, multiply by dydx\frac{dy}{dx}.
  3. Gather all dydx\frac{dy}{dx} terms to one side and factor out.
Worked Example: Find dydx\frac{dy}{dx} for x2+y2=1x^2 + y^2 = 1.
  • Step 1: 2x+2ydydx=02x + 2y\frac{dy}{dx} = 0.
  • Step 2: 2ydydx=−2x2y\frac{dy}{dx} = -2x.
  • Result: dydx=−xy\frac{dy}{dx} = -\frac{x}{y}.
If the exam gives an equation you cannot rearrange easily, implicit differentiation is always valid. Always group
dy/dxdy/dx terms first before dividing.

5. Tangent and Normal Lines

Tangent at (a,f(a))(a, f(a)): y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a)
Normal at (a,f(a))(a, f(a)): y−f(a)=−1f′(a)(x−a)y - f(a) = -\frac{1}{f'(a)}(x - a)

Abstract Solution (Strategy) — Tangent Problems

  1. [Find the point]: Evaluate f(a)f(a).
  2. [Find the slope]: Compute f′(a)f'(a).
  3. [Write the line]: Use the point-slope formula.

Worked Example

Question: Find the equation of the tangent to y=x3−2xy = x^3 - 2x at x=2x = 2.
  • Step 1: f(2)=8−4=4f(2) = 8 - 4 = 4. Point: (2,4)(2, 4).
  • Step 2: f′(x)=3x2−2  ⟹  f′(2)=10f'(x) = 3x^2 - 2 \implies f'(2) = 10.
  • Step 3: y−4=10(x−2)  ⟹  y=10x−16y - 4 = 10(x - 2) \implies y = 10x - 16.
  • Result: y=10x−16y = 10x - 16.

6. Functional Equations & Derivatives (Week 8 Link)

For the functional equation f(x+y)=f(x)f(y)f(x+y) = f(x)f(y), the function is exponential-type: f(x)=axf(x) = a^x.
  • Key property: f′(x)=f(x)⋅f′(0)f'(x) = f(x) \cdot f'(0).
  • Direct consequence: f′(1)=f(1)⋅f′(0)f'(1) = f(1) \cdot f'(0).
This is tested in Week 8, Questions 17 and 18.

7. Common Mistakes

MistakeWhy it happensCorrect approach
Power rule on exe^xTreating exe^x like xnx^n.ddx(ex)=ex\frac{d}{dx}(e^x) = e^x, not xex−1xe^{x-1}.
Wrong numerator order in Quotient RuleForgetting the formula direction.Always: u′v−uv′u'v - uv', never uv′−u′vuv' - u'v.
Missing the chain rule factorDifferentiating the outer but forgetting the inner.After every "outer" derivative, immediately multiply by the derivative of what's inside.
Assuming continuity = differentiabilityThe converse is false.$

8. Flashcards

<Flashcard front="State the Product Rule." back="(uv)' = u'v + uv'. The first derivative times the second, plus the first times the second derivative." /> <Flashcard front="State the Quotient Rule." back="(u/v)' = (u'v - uv') / v². Top prime first, minus sign in middle, always divide by the denominator squared." /> <Flashcard front="How do you differentiate f(g(x))?" back="Chain Rule: f'(g(x)) · g'(x). Outer derivative (inner unchanged) × inner derivative." /> <Flashcard front="Differentiability implies...?" back="Continuity. But continuity does NOT imply differentiability (counterexample: |x| at x=0)." /> <Flashcard front="What is d/dx(ln x)?" back="1/x. Valid only for x > 0." />

9. Practice Targets

  • Differentiate 5 functions using the Chain Rule (mix of trig + log + exponential).
  • Find the tangent and normal to y=xy = \sqrt{x} at x=4x = 4.
  • Use implicit differentiation on x3+y3=6xyx^3 + y^3 = 6xy to find dy/dxdy/dx.
  • Attempt Questions 6, 7, and 8 from Graded Assignment 8 (they are all derivative-based).

10. Connections

Connects toHow
Week 5 — ApplicationsEvery extremum, tangent, and optimization problem uses the derivatives built here.
Week 8 — Linear ApproxThe tangent line formula IS the linearization formula.
Statistics IMaximum Likelihood Estimation sets derivatives of log-likelihood to zero.

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