12.3PureCore

Differentiating xⁿ

First principles is slow. The power rule does the same job in one step: it is the workhorse of all A-Level differentiation. Multiply by the power, then drop the power by one.

25 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.3 Differentiating x^n (Polynomials)Watch the full walkthrough before the notes below.

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What you'll be able to do

State and apply the power rule
Differentiate terms with a coefficient
Differentiate negative and fractional powers
Rewrite terms into power form before differentiating

The power rule

Bring the power down to the front as a multiplier, then reduce the power by $1$ .

y = x^{n} \Rightarrow \frac{d y}{d x} = n x^{n - 1}

Multiply by the power, subtract one from the power.

1Multiply by 5, reduce power to 4.

Answer

\frac{d y}{d x} = 5 x^{4}

Coefficients

A constant multiplier just rides along: differentiate the power part and keep the coefficient.

1Power rule on

x^{4}

gives

4 x^{3}

2Keep the 3:

3 \times 4 x^{3}

Answer

12 x^{3}

Tip — Differentiating a plain number (like 7) gives 0 — constants have zero gradient.

Negative and fractional powers

The rule works for any power. Rewrite roots and reciprocals as powers first (e.g. $x = x^{1/2}$ , $\frac{1}{x} = x^{- 1}$ ), then apply the rule.

1Rewrite:

y = x^{1/2}

2Power rule:

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

Answer

\frac{1}{2} x^{- 1/2} = \frac{1}{2 x}

Tip — Always convert to index form before differentiating — you cannot apply the power rule to √x or 1/x directly.

Formula recap

y = x^{n} \Rightarrow \frac{d y}{d x} = n x^{n - 1}

The power rule.

y = a x^{n} \Rightarrow \frac{d y}{d x} = an x^{n - 1}

With a coefficient.

\frac{d}{d x} (constant) = 0

Constants differentiate to 0.

Common mistakes to avoid

Forgetting to reduce the power, e.g. d/dx(x³) = 3x³.

d/dx(x³) = 3x². Multiply by the power AND drop it by one.

Trying to differentiate √x without rewriting it.

Convert to x^(1/2) first, then apply the rule.

Key takeaways

Power rule: multiply by the power, reduce the power by 1.
Coefficients are carried through; constants differentiate to 0.
Rewrite roots/reciprocals as powers before differentiating.

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