Mathematics Advanced • Year 11 • Module 3 • Lesson 4

Differentiation Rules

Build procedural fluency with the power rule, constant-multiple rule and sum/difference rule — including rewriting radicals and reciprocals as powers first.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the power rule (valid for all real n):

d/dx ( xⁿ ) = ____ · x^____ .

Q1.2 Rewrite each expression as a single power of x (i.e. in the form xⁿ):

√x = x^____ 1 / x² = x^____ 1 / √x = x^____

Q1.3 Differentiate by rule (no first principles):

(a) d/dx (7) = ________ (b) d/dx (x) = ________ (c) d/dx (x²) = ________

Stuck? Revisit lesson § Power Rule and § Key Terms.

2. Worked example — f ′(x) for f(x) = 3x⁴ − 2x³ + 5x − 7

Follow each line of algebra. Every step has a reason on the right.

Problem. Differentiate f(x) = 3x⁴ − 2x³ + 5x − 7 using the power, constant-multiple and sum/difference rules.

Step 1 — Differentiate each term using the power rule.

d/dx (3x⁴) = 3 · 4 · x³ = 12x³

d/dx (−2x³) = −2 · 3 · x² = −6x²

d/dx (5x) = 5 · 1 · x⁰ = 5

d/dx (−7) = 0

Reason: power rule on each term; constant multiple slides past; derivative of a constant is 0.

Step 2 — Combine using the sum/difference rule.

f ′(x) = 12x³ − 6x² + 5

Reason: differentiation distributes over sums and differences.

Conclusion. f ′(x) = 12x³ − 6x² + 5.

3. Faded example — fill in the missing steps

Differentiate f(x) = √x + 1/x² . Fill in each blank. 3 marks

Step 1 — Rewrite in index notation:

f(x) = x^____ + x^____

Step 2 — Apply the power rule to each term:

d/dx ( x^1/2 ) = ____ · x^____

d/dx ( x⁻² ) = ____ · x^____

Step 3 — Combine (optionally rewrite without negative or fractional exponents):

f ′(x) = ________________ + ________________ (or) 1 / (2 √x ) − 2 / x³

Stuck? Revisit lesson § Worked Example 2 — rewrite as powers first.

4. Graduated practice — differentiate each function

Use the power, constant-multiple and sum/difference rules. Rewrite radicals/reciprocals as powers first when needed.

Foundation — power rule on monomials (4 questions)

Q	Function	f ′(x)
4.1 1	f(x) = x⁵
4.2 1	f(x) = 4x³
4.3 1	f(x) = 6x
4.4 1	f(x) = 9

Standard — typical HSC difficulty (6 questions)

Show the term-by-term derivative line, then combine.

4.5 f(x) = 4x³ − 3x² + 2x − 1. 2 marks

4.6 f(x) = 1 / x³ (rewrite first). 2 marks

4.7 f(x) = 3 √x − 2/x (rewrite first). 2 marks

4.8 Find f ′(1) for f(x) = 2x⁴ − 3x + 5. 2 marks

4.9 Find the gradient of the tangent to y = x³ − 2x² + 4 at x = −1. 2 marks

4.10 f(x) = (x² + 1)(x − 2) (expand first). 2 marks

Extension — combine concepts (2 questions)

4.11 Differentiate f(x) = x⁴ − 3x² + 2/x − √x . Show each rewrite, each derivative, and combine into a single expression. 3 marks

4.12 Find the x-coordinates of all points on the curve y = x³ − 3x at which the tangent is horizontal (gradient zero). 3 marks

Stuck on 4.12? Differentiate, set dy/dx = 0, then solve the resulting quadratic.

5. Self-check the easy 3

Tick the first three once you have checked your method works.

For 4.1 I subtracted 1 from the exponent (5 → 4) and brought the 5 down as a coefficient — both moves applied.

For 4.4 (constant function) I wrote 0 — not 9 — because the derivative of a constant is always 0.

For 4.6 I rewrote 1/x³ as x⁻³ before applying the power rule, then converted the answer back if asked.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Power rule

d/dx ( xⁿ ) = n · x^{n − 1} . Valid for any real number n.

Q1.2 — Index form

√x = x^1/2 . 1/x² = x⁻² . 1/√x = x^−1/2 .

Q1.3 — Simple derivatives

(a) d/dx (7) = 0. (b) d/dx (x) = 1. (c) d/dx (x²) = 2x.

Q3 — Faded example f(x) = √x + 1/x²

Step 1: f(x) = x^1/2 + x⁻². Step 2: d/dx (x^1/2) = (1/2) x^−1/2; d/dx (x⁻²) = −2 x⁻³. Step 3: f ′(x) = (1/2) x^−1/2 − 2 x⁻³ = 1 / (2 √x) − 2 / x³.

Q4.1 — f(x) = x⁵

f ′(x) = 5x⁴.

Q4.2 — f(x) = 4x³

f ′(x) = 4 · 3 x² = 12x².

Q4.3 — f(x) = 6x

f ′(x) = 6 (the derivative of x is 1; constant multiple 6 slides through).

Q4.4 — f(x) = 9

f ′(x) = 0 (constant function).

Q4.5 — f(x) = 4x³ − 3x² + 2x − 1

f ′(x) = 12x² − 6x + 2 − 0 = 12x² − 6x + 2.

Q4.6 — f(x) = 1/x³

Rewrite: x⁻³. f ′(x) = −3 x⁻⁴ = −3 / x⁴.

Q4.7 — f(x) = 3 √x − 2/x

Rewrite: 3 x^1/2 − 2 x⁻¹. Differentiate: 3 · (1/2) x^−1/2 − 2 · (−1) x⁻² = (3/2) x^−1/2 + 2 x⁻² = 3 / (2 √x) + 2 / x².

Q4.8 — f ′(1) for f(x) = 2x⁴ − 3x + 5

f ′(x) = 8x³ − 3. f ′(1) = 8 − 3 = 5.

Q4.9 — Gradient of y = x³ − 2x² + 4 at x = −1

dy/dx = 3x² − 4x. At x = −1: 3(1) − 4(−1) = 3 + 4 = 7.

Q4.10 — f(x) = (x² + 1)(x − 2)

Expand: x³ − 2x² + x − 2. f ′(x) = 3x² − 4x + 1.

Q4.11 — f(x) = x⁴ − 3x² + 2/x − √x

Rewrite: x⁴ − 3x² + 2 x⁻¹ − x^1/2. Differentiate term by term: 4x³, −6x, −2 x⁻², −(1/2) x^−1/2. Combine: f ′(x) = 4x³ − 6x − 2/x² − 1 / (2 √x).

Q4.12 — Horizontal tangents of y = x³ − 3x

dy/dx = 3x² − 3. Set = 0: 3x² = 3 ⇒ x² = 1 ⇒ x = 1 or x = −1. (Tangent is horizontal at the two stationary points of the cubic.)