12.5PureCore

Differentiating Functions with Two or More Terms

Differentiation is term-by-term, so any polynomial — however long — is handled by applying the power rule to each piece. The trick with harder expressions is preparing them into power form first.

25 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.5 Differentiating Functions with Two or More TermsWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Differentiate polynomials with several terms
Rewrite fractions and roots into power form first
Expand brackets before differentiating
Simplify the resulting gradient function

Term by term

Differentiate each term separately and add the results. The power rule applies to every term independently.

4 x^{3} \to 12 x^{2}

;

- 2 x^{2} \to - 4 x

;

7 x \to 7

;

- 9 \to 0

Answer

12 x^{2} - 4 x + 7

Prepare first: fractions and roots

You can only apply the power rule to powers of $x$ . Split fractions over a single denominator into separate power terms, and rewrite roots as fractional powers, before differentiating.

\frac{x ^{3} + 2 x}{x} = x^{2} + 2 \Rightarrow \frac{d y}{d x} = 2 x

Simplify into power form, then differentiate.

1Rewrite:

y = x^{- 2} + x^{1/2}

2Differentiate:

- 2 x^{- 3} + \frac{1}{2} x^{- 1/2}

Answer

- 2 x^{- 3} + \frac{1}{2} x^{- 1/2}

Tip — Get everything into the form axⁿ before differentiating — never differentiate a fraction or root directly.

Expand brackets first

If the function is a product of brackets, expand it into a polynomial before differentiating — the power rule does not apply to a product directly (at this stage).

1Expand:

y = x^{2} + 3 x

2Differentiate:

2 x + 3

Answer

2 x + 3

Formula recap

\frac{d}{d x} (u + v) = \frac{d u}{d x} + \frac{d v}{d x}

Differentiate term by term.

\frac{1}{x ^{n}} = x^{- n}, m x = x^{1/ m}

Rewrite before differentiating.

expand brackets first

No product rule yet.

Common mistakes to avoid

Differentiating a fraction like 1/x² as it stands.

Rewrite as x⁻² first, then apply the power rule.

Differentiating x(x + 3) without expanding.

Expand to x² + 3x first.

Key takeaways

Differentiate each term separately and add.
Rewrite fractions and roots into axⁿ form before differentiating.
Expand brackets first — there is no product rule at this stage.

Test yourself

Ready to lock in Differentiating Functions with Two or More Terms? Pick a mode and earn XP & Dobloons.