Maths › Differentiation › Differentiating powers of x

Differentiating powers of x

First principles proves the pattern; this lesson industrialises it. One rule differentiates every power of x, whole, negative or fractional, and with sums and constant multiples it handles any polynomial in sight, provided the expression is first rewritten into powers the rule can see.

Year 12-13EDEXCEL 9MA0 7.2

Builds on The derivative from first principles and Indices and surds.

IN THIS TOPIC

  • Differentiate xn for any rational n, with sums, differences and constant multiples.
  • Rewrite roots, reciprocals, products and quotients as powers before differentiating.

COMMON MISCONCEPTION

To differentiate a product, differentiate each factor.

The power rule

First principles delivered 2x from x2 and 3x2 from x3. The pattern those two suggest holds for every rational power, and it is on the must-learn list.

y=xn⇒dydx=nxn−1y = x^{n} \, ⇒ \, \frac{dy}{dx} = nx^{n−1}NOT IN THE BOOKLET — LEARN IT

Two companions turn it into an assembly line. A constant multiple simply rides along in front, and sums and differences are differentiated term by term. A constant on its own goes to zero, having no slope to report.

Tangent segments on y equals x squared at four points: the slope at every point is twice the x value, so the gradient is itself the function 2xslope −2slope 0slope 2slope 3the slope readout is itself a function: 2x
FIG. 1The rule, seen: tangent slopes on y = x² at four points read −2, 0, 2 and 3, always twice the x-value. The slope readout is the function 2x.

WORKED EXAMPLE

Three terms, three exponent styles

Differentiate y = 4x3 − 6√x + 2/x2.

Rewrite everything as powers first: y = 4x3 − 6x1/2 + 2x−2.

Apply the rule termwise: dy/dx = 12x2 − 3x−1/2 − 4x−3.

In the original notation, dy/dx = 12x2 − 3/√x − 4/x3.

The rewriting line is where the marks are awarded. Fractional and negative exponents obey the same rule as whole ones, but only once they are visibly exponents.

The curve y equals the square root of x with its tangent at 4 comma 2: the slope is one quarter, exactly what the power rule givesy = √xslope ¼ at (4, 2)fractional powers obey the same rule
FIG. 2The rule on a fractional power: y = √x is x to the ½, so its tangent at (4, 2) has slope ¼, exactly ½ times 4 to the power −½, drawn and computed.

Rewrite, then differentiate

The rule sees powers of x and nothing else. Products get expanded, fractions get split term by term, and only then does any differentiating happen. Differentiating each factor of (2x + 5)(x − 1) would hand you 2 × 1 = 2, a constant, at every point on the curve. Expand first and the truth looks nothing like it.

GUIDED PRACTICE

Expand first

Differentiate y = (2x + 5)(x − 1), before opening the working.

Show the working

Expand: y = 2x2 + 3x − 5.

Differentiate termwise: dy/dx = 4x + 3.

At x = 1 that is 7, while the factor-by-factor fake gives 2 everywhere. One numeric spot-check settles it, and a correct rule for products arrives in Year 13.

INDEPENDENT PRACTICE

Split the fraction

Differentiate y = (x2 + 3x − 5)/(4√x).

Show the working

Divide each term by 4x1/2: y = ¼x3/2 + ¾x1/2 − (5/4)x−1/2.

Differentiate termwise: dy/dx = (3/8)x1/2 + (3/8)x−1/2 + (5/8)x−3/2.

Tidied, dy/dx = (3√x)/8 + 3/(8√x) + 5/(8x√x).

No quotient was ever differentiated here. Only powers were. Reshaping an expression until the power rule applies is the whole craft at this stage.

ASSESSMENT FOCUS

  • Rewrite before differentiating. Every root, reciprocal and quotient becomes x to a power, and that rewriting line is usually a mark of its own.
  • Apply nxn−1 with the sign of n kept. The derivative of x−2 is −2x−3, the negative exponent growing more negative.
  • Give the final answer in the notation the question used. Convert x−1/2 back to 1/√x where the question was written in surds.

CHECK YOURSELF

Differentiate y = x4 − 2/√x + 7, and evaluate the gradient at x = 1.

Show a hint

Rewrite 2/√x as a power first; the 7 contributes nothing.

Show the answer

Rewritten, y = x4 − 2x−1/2 + 7, so dy/dx = 4x3 + x−3/2.

That is 4x3 + 1/(x√x) in surd form.

At x = 1 the gradient is 4 + 1 = 5, and the constant 7 never appeared, exactly as it should not.

Multiply down by the exponent, then knock the exponent down by one: nx to the n minus 1.

The rule sees only powers, so rewrite products, roots and fractions until powers are all there is.

WORKBOOK

Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.

Question workbook

PDF · 30 KB · print edition · white background

Question workbook

PDF · 30 KB · screen edition · navy

Answer book

PDF · 34 KB · fully worked · navy
7 questions on this topicAnswer them one at a time and mark yourself against the worked answer.Practise this topic

Or read them with their worked answers on the differentiating powers of x questions page.

CHECK YOUR PROGRESS

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  • Differentiate xn for any rational n, with sums, differences and constant multiples.
  • Rewrite roots, reciprocals, products and quotients as powers before differentiating.

Open the full revision checklist to see every objective in the course in one place.