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Differentiating xⁿ

Q: Does the power rule work for negative or fractional powers?

Yes — the power rule ddx(x^n)=nx^n-1 holds for any real number n, not just positive integers. Roots and reciprocals must first be rewritten as powers.

Q: Why does a constant term differentiate to zero?

A constant has no x in it, so it represents a horizontal line — which has zero gradient everywhere. Differentiating it correctly gives zero.

First principles proved that differentiating $x^{2}$ gives $2 x$ and $x^{3}$ gives $3 x^{2}$ . The power rule packages that pattern into a single formula you can apply instantly to ANY power — positive, negative or fractional — without ever writing out a limit again.

26 min Video by TLMaths Differentiation

The big picture

The power rule is the workhorse of the entire calculus course: bring the power down as a multiplier, then reduce the power by one. It applies to every real-number power, not just positive integers, which means roots and reciprocals can be differentiated too, once they are rewritten as fractional or negative powers. Every later differentiation technique (chain, product, quotient) is really the power rule applied inside a more complicated structure.

A-Level Maths G2-01: Differentiate x^n, n a positive integerWatch the walkthrough, then work through the notes below.

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What you'll be able to do

State and apply the power rule for any real index $n$
Differentiate sums of power terms term-by-term
Rewrite roots and reciprocals as powers before differentiating
Differentiate expressions after expanding brackets

The power rule

For $y = x^{n}$ , where $n$ is any real number, the derivative is found by bringing the power down as a coefficient and reducing the power by 1. This holds for positive integers (proved by first principles), negative integers, and fractional powers alike.

\frac{d}{d x} (x^{n}) = n x^{n - 1}

Valid for any real number n.

1Bring the power down: coefficient 5.

2Reduce the power by 1:

x^{4}

Answer

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

Tip — Rewrite roots as fractional powers before differentiating: $x = x^{1/2}$ , $3 x = x^{1/3}$ — then the power rule applies directly.

A constant coefficient

If a power term has a constant multiplier, that constant stays put and multiplies the result — you differentiate the $x^{n}$ part and keep the constant factor.

\frac{d}{d x} (a x^{n}) = an x^{n - 1}

The constant a is carried through unchanged.

1Differentiate

x^{3}

3 x^{2}

2Multiply by the constant 4:

4 \times 3 x^{2}

Answer

\frac{d y}{d x} = 12 x^{2}

Term-by-term for polynomials

A sum of power terms is differentiated one term at a time — each term uses the power rule independently, and a constant term (with no $x$ ) always differentiates to zero, since it has no gradient.

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

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

5 x \to 5

(since

5 x^{1} \to 5 \times 1 \times x^{0} = 5

- 7 \to 0

(a constant has zero gradient).

Answer

\frac{d y}{d x} = 12 x^{3} - 4 x + 5

Why this works

This term-by-term approach — differentiate each piece separately and add — works because differentiation is linear: the derivative of a sum is the sum of the derivatives.

Think like an examiner

Always rewrite roots and fractions as powers of $x$ FIRST — trying to differentiate $x$ or $\frac{1}{x ^{2}}$ directly without rewriting is a common source of errors.
Remember a plain constant term always differentiates to 0 — it is easy to forget it entirely rather than writing the zero.
If brackets are given (e.g. $(x + 1) (x - 2)$ ), expand fully into a polynomial BEFORE differentiating term-by-term — the power rule alone does not apply to a product directly.

Common misconceptions

Differentiating

x

as if it stays

x

multiplied by something.

Rewrite

x = x^{1/2}

first, then apply the power rule to get

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

Forgetting a constant term differentiates to zero and instead copying it unchanged.

A term with no

x

(like

- 7

) has zero gradient — it vanishes entirely when you differentiate.

Applying the power rule directly to an unexpanded product like

(x + 3) (x - 1)

Expand into a polynomial first (e.g.

x^{2} + 2 x - 3

), THEN differentiate term-by-term.

The power rule

\frac{d}{d x} (x^{n}) = n x^{n - 1}

For any real n.

\frac{d}{d x} (a x^{n}) = an x^{n - 1}

Constant coefficient carried through.

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

A constant has zero derivative.

Stretch yourself

Find $\frac{d y}{d x}$ for $y = \frac{( x + 2 ) ( x - 3 )}{x}$ , fully simplifying your answer.

Hint — Expand the numerator first, then divide every term by $x$ to write $y$ as a sum of powers of $x$ , before differentiating term-by-term.

Questions students ask

Does the power rule work for negative or fractional powers?

Yes — the power rule

\frac{d}{d x} (x^{n}) = n x^{n - 1}

holds for any real number

n

, not just positive integers. Roots and reciprocals must first be rewritten as powers.

Why does a constant term differentiate to zero?

A constant has no

x

in it, so it represents a horizontal line — which has zero gradient everywhere. Differentiating it correctly gives zero.

Can I differentiate a product of two brackets directly with the power rule?

Not directly — you must expand the brackets into a single polynomial first. (Later, the product rule handles products WITHOUT expanding, which is essential when expanding isn't practical.)

Key takeaways

Power rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ , valid for any real $n$ .
A constant coefficient is carried through unchanged.
A polynomial is differentiated term-by-term; constant terms vanish to zero.
Rewrite roots and reciprocals as powers of $x$ before differentiating.

How this fits the course

Build on

Tangents and Normals

Leads to

Tangents and Normals

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