Second derivatives and curve behaviour Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Second derivatives and curve behaviour questions. See exactly how to solve problems on second derivative, concavity.

second derivativeconcavity
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The curve has equation y=x2−4x+7y=x^{2} - 4 x + 7. Find d2ydx2\frac{d^2y}{dx^2}.

Worked solution

  1. Differentiate to find the first derivative

    dydx=2x−4\frac{dy}{dx}=2 x - 4

    Differentiate the function term by term.

  2. Differentiate the first derivative again

    ddx(2x−4)=2\frac{d}{dx}\left(2 x - 4\right)=2

    Differentiating the gradient function gives the second derivative.

  3. State the second derivative

    d2ydx2=2\frac{d^2y}{dx^2}=2

    This is the required second derivative.

Answer
22
Question 2
2 markseasy
The curve has equation y=x3−2xy=x^{3} - 2 x. Find d2ydx2\frac{d^2y}{dx^2}.

Worked solution

  1. Differentiate to find the first derivative

    dydx=3x2−2\frac{dy}{dx}=3 x^{2} - 2

    Differentiate the function term by term.

  2. Differentiate the first derivative again

    ddx(3x2−2)=6x\frac{d}{dx}\left(3 x^{2} - 2\right)=6 x

    Differentiating the gradient function gives the second derivative.

  3. State the second derivative

    d2ydx2=6x\frac{d^2y}{dx^2}=6 x

    This is the required second derivative.

Answer
6x6 x
Question 3
2 markseasy
The curve has equation y=x3+x2y=x^{3} + x^{2}. Find d2ydx2\frac{d^2y}{dx^2}.

Worked solution

  1. Differentiate to find the first derivative

    dydx=3x2+2x\frac{dy}{dx}=3 x^{2} + 2 x

    Differentiate the function term by term.

  2. Differentiate the first derivative again

    ddx(3x2+2x)=2(3x+1)\frac{d}{dx}\left(3 x^{2} + 2 x\right)=2 \left(3 x + 1\right)

    Differentiating the gradient function gives the second derivative.

  3. State the second derivative

    d2ydx2=2(3x+1)\frac{d^2y}{dx^2}=2 \left(3 x + 1\right)

    This is the required second derivative.

Answer
2(3x+1)2 \left(3 x + 1\right)
Question 4
2 markseasy
The curve has equation y=x4y=x^{4}. Find d2ydx2\frac{d^2y}{dx^2}.

Worked solution

  1. Differentiate to find the first derivative

    dydx=4x3\frac{dy}{dx}=4 x^{3}

    Differentiate the function term by term.

  2. Differentiate the first derivative again

    ddx(4x3)=12x2\frac{d}{dx}\left(4 x^{3}\right)=12 x^{2}

    Differentiating the gradient function gives the second derivative.

  3. State the second derivative

    d2ydx2=12x2\frac{d^2y}{dx^2}=12 x^{2}

    This is the required second derivative.

Answer
12x212 x^{2}
Question 5
2 markseasy
The curve has equation y=2x3−xy=2 x^{3} - x. Find d2ydx2\frac{d^2y}{dx^2}.

Worked solution

  1. Differentiate to find the first derivative

    dydx=6x2−1\frac{dy}{dx}=6 x^{2} - 1

    Differentiate the function term by term.

  2. Differentiate the first derivative again

    ddx(6x2−1)=12x\frac{d}{dx}\left(6 x^{2} - 1\right)=12 x

    Differentiating the gradient function gives the second derivative.

  3. State the second derivative

    d2ydx2=12x\frac{d^2y}{dx^2}=12 x

    This is the required second derivative.

Answer
12x12 x

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