Standard derivatives Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Standard derivatives questions. See exactly how to solve problems on differentiation, standard derivatives.

differentiationstandard derivatives
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The curve has equation y=x5y=x^{5}. Find dydx\frac{dy}{dx}.

Worked solution

  1. Recall the power rule

    ddx(a xn)=a n xn−1\frac{d}{dx}\left(a\,x^{n}\right)=a\,n\,x^{n-1}

    Multiply by the power, then reduce the power by one.

  2. Differentiate the function

    ddx(x5)=5x4\frac{d}{dx}\left(x^{5}\right)=5 x^{4}

    Apply the standard result to the function.

  3. State the derivative

    dydx=5x4\boxed{\dfrac{dy}{dx}=5 x^{4}}

    This is the gradient function of the curve.

Answer
5x45 x^{4}
Question 2
2 markseasy
The curve has equation y=3x4y=3 x^{4}. Find dydx\frac{dy}{dx}.

Worked solution

  1. Recall the power rule

    ddx(a xn)=a n xn−1\frac{d}{dx}\left(a\,x^{n}\right)=a\,n\,x^{n-1}

    Multiply by the power, then reduce the power by one.

  2. Differentiate the function

    ddx(3x4)=12x3\frac{d}{dx}\left(3 x^{4}\right)=12 x^{3}

    Apply the standard result to the function.

  3. State the derivative

    dydx=12x3\boxed{\dfrac{dy}{dx}=12 x^{3}}

    This is the gradient function of the curve.

Answer
12x312 x^{3}
Question 3
2 markseasy
The curve has equation y=1x3y=\frac{1}{x^{3}}. Find dydx\frac{dy}{dx}.

Worked solution

  1. Recall the power rule

    ddx(a xn)=a n xn−1\frac{d}{dx}\left(a\,x^{n}\right)=a\,n\,x^{n-1}

    Multiply by the power, then reduce the power by one.

  2. Differentiate the function

    ddx(1x3)=−3x4\frac{d}{dx}\left(\frac{1}{x^{3}}\right)=- \frac{3}{x^{4}}

    Apply the standard result to the function.

  3. State the derivative

    dydx=−3x4\boxed{\dfrac{dy}{dx}=- \frac{3}{x^{4}}}

    This is the gradient function of the curve.

Answer
−3x4- \frac{3}{x^{4}}
Question 4
2 markseasy
The curve has equation y=6xy=6 \sqrt{x}. Find dydx\frac{dy}{dx}.

Worked solution

  1. Recall the power rule

    ddx(a xn)=a n xn−1\frac{d}{dx}\left(a\,x^{n}\right)=a\,n\,x^{n-1}

    Multiply by the power, then reduce the power by one.

  2. Differentiate the function

    ddx(6x)=3x\frac{d}{dx}\left(6 \sqrt{x}\right)=\frac{3}{\sqrt{x}}

    Apply the standard result to the function.

  3. State the derivative

    dydx=3x\boxed{\dfrac{dy}{dx}=\frac{3}{\sqrt{x}}}

    This is the gradient function of the curve.

Answer
3x\frac{3}{\sqrt{x}}
Question 5
2 markseasy
The curve has equation y=e3xy=e^{3 x}. Find dydx\frac{dy}{dx}.

Worked solution

  1. Recall the derivative of the exponential

    ddx(ekx)=k ekx\frac{d}{dx}\left(e^{kx}\right)=k\,e^{kx}

    The exponential differentiates to itself, times the constant k.

  2. Differentiate the function

    ddx(e3x)=3e3x\frac{d}{dx}\left(e^{3 x}\right)=3 e^{3 x}

    Apply the standard result to the function.

  3. State the derivative

    dydx=3e3x\boxed{\dfrac{dy}{dx}=3 e^{3 x}}

    This is the gradient function of the curve.

Answer
3e3x3 e^{3 x}

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