Differentiating powers and polynomials | Year 11

01 / LEARN

The idea, step by step

The power rule

For a positive integer power, multiply by the old exponent and reduce that exponent by one. Retain any coefficient as a multiplier. A constant has no change, so its derivative is zero.

Apply the rule separately to each term. Remember x means 1x¹: its derivative is 1. Losing a coefficient or changing only the exponent gives the wrong gradient function.

\frac{d}{d x} (a x^{n}) = a n x^{n - 1} (n \geq 1), \frac{d}{d x} (c) = 0.

Simplify before differentiating

Expand products or collect like terms before applying the power rule. For suitable polynomial expressions this avoids needing later rules for products and quotients. Keep the same independent variable throughout.

The derivative of a nonconstant polynomial has degree one less. As a check, compare a derivative value with nearby secant slopes or inspect whether the resulting degree and signs are plausible.

A common trap

Reducing the exponent but forgetting to multiply by it.

Instead: Both steps matter: $8 x^{3}$ differentiates to $24 x^{2}$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Differentiate $f (x) = 4 x^{3} - 5 x^{2} + 7 x - 9$ .

The term derivatives are $12 x^{2}$ , $- 10 x$ , 7 and 0.
Combine the terms with their original signs.

f^{'} (x) = 12 x^{2} - 10 x + 7

Example 2

Worked example 2

Differentiate $y = (x + 2) (x - 3)$ .

Expand to $y = x^{2} - x - 6$ .
Apply the power rule to each term.

d y / d x = 2 x - 1

Check your understanding

Differentiate $7 x^{4} - 2 x$ .

Show answer

$28 x^{3} - 2$

The derivatives are $7 (4) x^{3}$ and −2.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Differentiate $3 x^{4} - 2 x^{2} + 5$ .
3 marks
Find d/dt of $6 t^{3} + t - 8$ .
3 marks
Differentiate $y = 9 - 4 x$ .
3 marks
Differentiate $(x + 2) (x - 3)$ .
3 marks
Differentiate $f (x) = \frac{1}{2} x^{4} - 3 x^{2} + 6 x$ .
3 marks
A student differentiates $8 x^{3}$ as $8 x^{2}$ . Correct and explain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Differentiate $3 x^{4} - 2 x^{2} + 5$ .
3 marks
Differentiate each power; the constant 5 contributes 0.
$12 x^{3} - 4 x$
Find d/dt of $6 t^{3} + t - 8$ .
3 marks
$6 (3) t^{2} + 1 - 0 = 18 t^{2} + 1$ .
$18 t^{2} + 1$
Differentiate $y = 9 - 4 x$ .
3 marks
The derivative of 9 is 0; that of −4x is −4.
−4
Differentiate $(x + 2) (x - 3)$ .
3 marks
Expand to $x^{2} - x - 6$ , then differentiate.
$2 x - 1$
Differentiate $f (x) = \frac{1}{2} x^{4} - 3 x^{2} + 6 x$ .
3 marks
$\frac{1}{2} (4) x^{3} - 3 (2) x + 6 = 2 x^{3} - 6 x + 6$ .
$2 x^{3} - 6 x + 6$
A student differentiates $8 x^{3}$ as $8 x^{2}$ . Correct and explain.
3 marks
Multiply the coefficient by 3 while reducing the exponent to 2.
$24 x^{2}$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topic 6: rates of change, derivatives and applications. Introductory polynomial calculus with explicit modelling and interpretation.

Official curriculum source ↗

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