Sec 4 A-Math · Differentiation

The power rule questions with answers

Practice questions on the power rule, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Given that $y = 4x^{3} - 5x^{2} + x - 1$ , find $\dfrac{dy}{dx}$ .

Step 1 of 1

Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0)

\dfrac{dy}{dx} = 4(3)x^{2} - 5(2)x + 1

= 12x^{2} - 10x + 1

Try a question: The power rule

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Questions

Question 1Polynomials
Given that $y = 4x^{3} - 4x^{2} + 7x - 1$ , find $\dfrac{dy}{dx}$ .
Show the worked solution
1. Step 1: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 4(3)x^{2} - 4(2)x + 7(1)$ $= 12x^{2} - 8x + 7$
Question 2Polynomials
Given that $y = 5x^{3} + 6x^{2} + 7x - 1$ , find $\dfrac{dy}{dx}$ .
Show the worked solution
1. Step 1: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 5(3)x^{2} + 6(2)x + 7(1)$ $= 15x^{2} + 12x + 7$
Question 3Negative and fractional powers
Given that $y = 4x^{2} + \dfrac{9}{\sqrt[3]{x^{2}}}$ , find $\dfrac{dy}{dx}$ .
Show the worked solution
1. Step 1: Write every term as a power of x: 1 over xⁿ is x to the power −n, and a root is a fractional power $y = 4x^{2} + 9x^{-\frac{2}{3}}$
2. Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 4(2)x + 9(-\dfrac{2}{3})x^{-\frac{5}{3}}$ $= 8x - 6x^{-\frac{5}{3}}$
3. Step 3: Write the answer with positive indices $= 8x - \dfrac{6}{x^{\frac{5}{3}}}$
Question 4Negative and fractional powers
Given that $y = 6\sqrt{x} + \dfrac{4}{x}$ , find $\dfrac{dy}{dx}$ .
Show the worked solution
1. Step 1: Write every term as a power of x: 1 over xⁿ is x to the power −n, and a root is a fractional power $y = 6x^{\frac{1}{2}} + 4x^{-1}$
2. Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 6(\dfrac{1}{2})x^{-\frac{1}{2}} + 4(-1)x^{-2}$ $= 3x^{-\frac{1}{2}} - 4x^{-2}$
3. Step 3: Write the answer with positive indices $= \dfrac{3}{\sqrt{x}} - \dfrac{4}{x^{2}}$
Question 5Expand or split first
Given that $y = \dfrac{4x^{2} + x + 3}{x^{2}}$ , find $\dfrac{dy}{dx}$ .
Show the worked solution
1. Step 1: Split the fraction: divide each term on top by the bottom $y = 4 + x^{-1} + 3x^{-2}$
2. Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = -x^{-2} + 3(-2)x^{-3}$ $= -x^{-2} - 6x^{-3}$
3. Step 3: Write the answer with positive indices $= -\dfrac{1}{x^{2}} - \dfrac{6}{x^{3}}$
Question 6Expand or split first
Given that $y = \dfrac{x - 8}{\sqrt{x}}$ , find $\dfrac{dy}{dx}$ .
Show the worked solution
1. Step 1: Split the fraction: divide each term on top by the bottom $y = x^{\frac{1}{2}} - 8x^{-\frac{1}{2}}$
2. Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = \dfrac{1}{2}x^{-\frac{1}{2}} - 8(-\dfrac{1}{2})x^{-\frac{3}{2}}$ $= \dfrac{1}{2}x^{-\frac{1}{2}} + 4x^{-\frac{3}{2}}$
3. Step 3: Write the answer with positive indices $= \dfrac{1}{2\sqrt{x}} + \dfrac{4}{x\sqrt{x}}$

More on differentiation

Notes: Differentiation Worksheet with answers (PDF)

Endless questions on the power rule

A free account gives you new questions every time, step-by-step help, and tracks what you've mastered.

Start free

Worked example

Given that

y = 4x^{3} - 5x^{2} + x - 1

, find

\dfrac{dy}{dx}

Step 1 of 1

Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0)

\dfrac{dy}{dx} = 4(3)x^{2} - 5(2)x + 1

= 12x^{2} - 10x + 1

Questions

Question 1Polynomials

Given that $y = 4x^{3} - 4x^{2} + 7x - 1$ , find $\dfrac{dy}{dx}$ .

Show the worked solution

Step 1: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 4(3)x^{2} - 4(2)x + 7(1)$ $= 12x^{2} - 8x + 7$

Question 2Polynomials

Given that $y = 5x^{3} + 6x^{2} + 7x - 1$ , find $\dfrac{dy}{dx}$ .

Show the worked solution

Step 1: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 5(3)x^{2} + 6(2)x + 7(1)$ $= 15x^{2} + 12x + 7$

Question 3Negative and fractional powers

Given that $y = 4x^{2} + \dfrac{9}{\sqrt[3]{x^{2}}}$ , find $\dfrac{dy}{dx}$ .

Show the worked solution

Step 1: Write every term as a power of x: 1 over xⁿ is x to the power −n, and a root is a fractional power $y = 4x^{2} + 9x^{-\frac{2}{3}}$
Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 4(2)x + 9(-\dfrac{2}{3})x^{-\frac{5}{3}}$ $= 8x - 6x^{-\frac{5}{3}}$
Step 3: Write the answer with positive indices $= 8x - \dfrac{6}{x^{\frac{5}{3}}}$

Question 4Negative and fractional powers

Given that $y = 6\sqrt{x} + \dfrac{4}{x}$ , find $\dfrac{dy}{dx}$ .

Show the worked solution

Step 1: Write every term as a power of x: 1 over xⁿ is x to the power −n, and a root is a fractional power $y = 6x^{\frac{1}{2}} + 4x^{-1}$
Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = 6(\dfrac{1}{2})x^{-\frac{1}{2}} + 4(-1)x^{-2}$ $= 3x^{-\frac{1}{2}} - 4x^{-2}$
Step 3: Write the answer with positive indices $= \dfrac{3}{\sqrt{x}} - \dfrac{4}{x^{2}}$

Question 5Expand or split first

Given that $y = \dfrac{4x^{2} + x + 3}{x^{2}}$ , find $\dfrac{dy}{dx}$ .

Show the worked solution

Step 1: Split the fraction: divide each term on top by the bottom $y = 4 + x^{-1} + 3x^{-2}$
Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = -x^{-2} + 3(-2)x^{-3}$ $= -x^{-2} - 6x^{-3}$
Step 3: Write the answer with positive indices $= -\dfrac{1}{x^{2}} - \dfrac{6}{x^{3}}$

Question 6Expand or split first

Given that $y = \dfrac{x - 8}{\sqrt{x}}$ , find $\dfrac{dy}{dx}$ .

Show the worked solution

Step 1: Split the fraction: divide each term on top by the bottom $y = x^{\frac{1}{2}} - 8x^{-\frac{1}{2}}$
Step 2: Differentiate each term: bring the power down to multiply, then take 1 off the power (a constant gives 0) $\dfrac{dy}{dx} = \dfrac{1}{2}x^{-\frac{1}{2}} - 8(-\dfrac{1}{2})x^{-\frac{3}{2}}$ $= \dfrac{1}{2}x^{-\frac{1}{2}} + 4x^{-\frac{3}{2}}$
Step 3: Write the answer with positive indices $= \dfrac{1}{2\sqrt{x}} + \dfrac{4}{x\sqrt{x}}$