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Class 12 Statistics Notes · GSEB

Working Rules

Working Rules — master the sum, product, quotient and chain rules of differentiation with mnemonics and step-by-step illustrations. GSEB Class 12 Commerce Statistics notes.

Last updated: 24 Sep 2026

Notes

Working Rules of Differentiation

Working Rules
Four accepted rules — sum, product, quotient and chain — that let you differentiate compound expressions without returning to the limit definition every time. If u and v are differentiable functions of x, these rules always apply.

Why they work

Each rule is proved once from the limit definition; after that you simply reuse the result. In exams you state the rule and apply it — no proof needed at this level.

The Four Rules at a Glance

1 · Sum / Difference Rule

ddx(u±v)=dudx±dvdx\frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}

Differentiate each term on its own. The workhorse for polynomials.

2 · Product Rule

ddx(u⋅v)=udvdx+vdudx\frac{d}{dx}(u \cdot v) = u\frac{dv}{dx} + v\frac{du}{dx}

Two functions multiplied — never multiply their derivatives.

3 · Quotient Rule

ddx(uv)=vdudx−udvdxv2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}

Division, v ≠ 0. Watch the order: low-d-high minus high-d-low.

4 · Chain Rule

dydx=dydu×dudx\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}

Function inside a function — outside first, then × inside derivative.

The product-rule mistake that won't die

For y = u·v, students write u′·v′. Wrong. It is u·v′ + v·u′. Quick check: y = x · x = x², so derivative must be 2x. The rule gives x·1 + x·1 = 2x ✓ — while u′v′ gives 1 ✗.

Walk Through Each Rule

Four tabs, four real textbook illustrations. Click through each rule and follow the steps as they appear.

Rule Selector — click a rule, watch the worked example unfold

Product Rule

Formula

dydx=u⋅dvdx+v⋅dudx\frac{dy}{dx} = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}

(u·v)′ = u·v′ + v·u′

First × derivative of second + second × derivative of first.

Worked example: y = (2x² + 3)(3x − 2)

1

Split the factors

y = (2x² + 3)(3x − 2) → u = 2x² + 3, v = 3x − 2

2

Differentiate each factor

du/dx = 4x · dv/dx = 3

3

Apply u·v′ + v·u′

(2x² + 3)(3) + (3x − 2)(4x)

4

Expand and collect

6x² + 9 + 12x² − 8x = 18x² − 8x + 9

Result of worked example

dy/dx = 18x² − 8x + 9

Try it yourself — Product Rule

Q. Find dy/dx if y = (x² + 1)(2x − 3).

Hint: u = x² + 1, v = 2x − 3 → apply u·v′ + v·u′, then expand.

Answer

dy/dx = 6x² − 6x + 2

Choosing the Right Rule

1

Terms added or subtracted?

→ Sum rule — differentiate term by term

2

Two functions multiplied?

→ Product rule — u·v′ + v·u′

3

One function over another?

→ Quotient rule — (v·u′ − u·v′)/v²

4

Function nested inside another?

→ Chain rule — outside′ × inside′

Simplification first, rule second

y = (x + 6/(x+5))·((3x+2)/(x²+5x+6)) looks terrifying — but the textbook cancels it to (3x+2)/(x+5) first, then applies the quotient rule. Always simplify before differentiating.

Solved Examples

Solved Example

Problem

Find dy/dx if y = (2x² + 3)(3x − 2).

Solution

dy/dx = 18x² − 8x + 9

Solved Example

Problem

Find dy/dx if y = √(x² + 3).

Solution

dy/dx = x / √(x² + 3)

Solved Example

Problem

Find dy/dx if y = 3/(4x + 5).

Solution

dy/dx = −12 / (4x + 5)²

Mnemonics Wall

Product rule

“First times d-second, plus second times d-first.”

Quotient rule

“Low d-high minus high d-low, over low squared.”

Chain rule

“Outside in, then times the inside.”

Sum rule

“Split the terms, finish each term.”

Real-life negative example

A tiffin service priced its combo as (roti count) × (curry price) and tracked revenue wrong by multiplying the changes instead of applying the product rule logic. When roti count rose 10% and curry price rose 5%, revenue did not rise 0.5% — it rose about 15.5% (u·v′ + v·u′ structure). Multiplying derivatives is wrong in maths and in business.

Key Takeaways

Key Takeaways

  • Sum rule: (u ± v)′ = u′ ± v′ — differentiate term by term.
  • ★ Product rule: (u·v)′ = u·v′ + v·u′ — never u′·v′.
  • ★ Quotient rule: (u/v)′ = (v·u′ − u·v′)/v² — low d-high minus high d-low, over low squared.
  • ★ Chain rule: differentiate the outside, multiply by the derivative of the inside.
  • Simplify the expression first — the textbook routinely cancels before differentiating.
  • So what? These four rules turn into marginal revenue, marginal cost and elasticity formulas in the business topics ahead.

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