Rates of change Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Rates of change questions. See exactly how to solve problems on rates of change, differentiation, connected rates, chain rule.

rates of changedifferentiationconnected rateschain ruleexponential growthdifferential equation
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
A sphere of radius rr has volume V=4πr33V=\frac{4 \pi r^{3}}{3}. Find dVdr\frac{dV}{dr}, the rate of change of the quantity with respect to rr.

Worked solution

  1. Write down the given formula

    V=4πr33V=\frac{4 \pi r^{3}}{3}

    State the quantity as a function of the variable.

  2. Differentiate with respect to r

    ddr(4πr33)=4πr2\frac{d}{dr}\left(\frac{4 \pi r^{3}}{3}\right)=4 \pi r^{2}

    Apply the power rule to each term.

  3. State the derivative

    dVdr=4πr2\boxed{\frac{dV}{dr}=4 \pi r^{2}}

    This is the required rate of change.

Answer
4πr24 \pi r^{2}
Question 2
2 markseasy
A sphere of radius rr has surface area A=4πr2A=4 \pi r^{2}. Find dAdr\frac{dA}{dr}, the rate of change of the quantity with respect to rr.

Worked solution

  1. Write down the given formula

    A=4πr2A=4 \pi r^{2}

    State the quantity as a function of the variable.

  2. Differentiate with respect to r

    ddr(4πr2)=8πr\frac{d}{dr}\left(4 \pi r^{2}\right)=8 \pi r

    Apply the power rule to each term.

  3. State the derivative

    dAdr=8πr\boxed{\frac{dA}{dr}=8 \pi r}

    This is the required rate of change.

Answer
8πr8 \pi r
Question 3
2 markseasy
A circle of radius rr has area A=πr2A=\pi r^{2}. Find dAdr\frac{dA}{dr}, the rate of change of the quantity with respect to rr.

Worked solution

  1. Write down the given formula

    A=πr2A=\pi r^{2}

    State the quantity as a function of the variable.

  2. Differentiate with respect to r

    ddr(πr2)=2πr\frac{d}{dr}\left(\pi r^{2}\right)=2 \pi r

    Apply the power rule to each term.

  3. State the derivative

    dAdr=2πr\boxed{\frac{dA}{dr}=2 \pi r}

    This is the required rate of change.

Answer
2πr2 \pi r
Question 4
2 markseasy
A circle of radius rr has circumference C=2πrC=2 \pi r. Find dCdr\frac{dC}{dr}, the rate of change of the quantity with respect to rr.

Worked solution

  1. Write down the given formula

    C=2πrC=2 \pi r

    State the quantity as a function of the variable.

  2. Differentiate with respect to r

    ddr(2πr)=2π\frac{d}{dr}\left(2 \pi r\right)=2 \pi

    Apply the power rule to each term.

  3. State the derivative

    dCdr=2π\boxed{\frac{dC}{dr}=2 \pi}

    This is the required rate of change.

Answer
2π2 \pi
Question 5
2 markseasy
A cube of side xx has volume V=x3V=x^{3}. Find dVdx\frac{dV}{dx}, the rate of change of the quantity with respect to xx.

Worked solution

  1. Write down the given formula

    V=x3V=x^{3}

    State the quantity as a function of the variable.

  2. Differentiate with respect to x

    ddx(x3)=3x2\frac{d}{dx}\left(x^{3}\right)=3 x^{2}

    Apply the power rule to each term.

  3. State the derivative

    dVdx=3x2\boxed{\frac{dV}{dx}=3 x^{2}}

    This is the required rate of change.

Answer
3x23 x^{2}

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