A-Level Standard derivatives Practice Questions

Free A-Level Standard derivatives practice questions with full step-by-step worked solutions. Covers differentiation, standard derivatives. Practise exam-style problems and check your method.

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}.
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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
Which expression is the derivative of f(x)=4x3f(x)=4 x^{3} with respect to xx?
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Worked solution

  1. Differentiate the function

    dydx=12x2\frac{dy}{dx}=12 x^{2}

    Differentiate using the standard results.

  2. Compare with each option

    match against the five options\text{match against the five options}

    Look for the option equal to this derivative.

  3. Select the correct option

    12x212 x^{2}

    This matches the derivative found above.

Answer
12x212 x^{2}
Question 3
3 marksintermediate
Which expression is the derivative of f(x)=1x2f(x)=\frac{1}{x^{2}} with respect to xx?
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Worked solution

  1. Write down the function

    y=1x2y=\frac{1}{x^{2}}

    State clearly what is being differentiated.

  2. 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.

  3. Differentiate using the standard result

    ddx(1x2)=−2x3\frac{d}{dx}\left(\frac{1}{x^{2}}\right)=- \frac{2}{x^{3}}

    Apply the standard derivative to the term.

  4. Recall that differentiation is linear

    ddx(f+g)=dfdx+dgdx\frac{d}{dx}\left(f+g\right)=\frac{df}{dx}+\frac{dg}{dx}

    Differentiate each term separately and add the results.

  5. Compare the derivative with the options

    dydx=−2x3\frac{dy}{dx}=- \frac{2}{x^{3}}

    Choose the option equal to this expression.

  6. Select the correct option

    −2x3- \frac{2}{x^{3}}

    This matches the derivative found above.

Answer
−2x3- \frac{2}{x^{3}}
Question 4
5 markshard
Which expression is the derivative of f(x)=x32f(x)=x^{\frac{3}{2}} with respect to xx?
Show worked solution

Worked solution

  1. Write down the function

    y=x32y=x^{\frac{3}{2}}

    State clearly what is being differentiated.

  2. 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.

  3. Differentiate using the standard result

    ddx(x32)=3x2\frac{d}{dx}\left(x^{\frac{3}{2}}\right)=\frac{3 \sqrt{x}}{2}

    Apply the standard derivative to the term.

  4. Recall that differentiation is linear

    ddx(f+g)=dfdx+dgdx\frac{d}{dx}\left(f+g\right)=\frac{df}{dx}+\frac{dg}{dx}

    Differentiate each term separately and add the results.

  5. Note that constant multiples are preserved

    ddx(c f)=c dfdx\frac{d}{dx}\left(c\,f\right)=c\,\frac{df}{dx}

    A constant factor multiplies straight through the derivative.

  6. Remember that constants differentiate to zero

    ddx(c)=0\frac{d}{dx}\left(c\right)=0

    Any constant term disappears when differentiated.

  7. Confirm every term has been differentiated

    all 1 term(s) processed\text{all }1\text{ term(s) processed}

    No term should be left unchanged in the gradient function.

  8. Check each result against the formula book

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

    Standard derivatives are provided in the formula booklet.

  9. Compare the derivative with the options

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

    Choose the option equal to this expression.

  10. Select the correct option

    3x2\frac{3 \sqrt{x}}{2}

    This matches the derivative found above.

Answer
3x2\frac{3 \sqrt{x}}{2}
Question 5
8 markschallenging
Which expression is the derivative of f(x)=e3x+ln⁡(2x)f(x)=e^{3 x} + \ln{\left(2 x \right)} with respect to xx?
Show worked solution

Worked solution

  1. Write down the function

    y=e3x+ln⁡(2x)y=e^{3 x} + \ln{\left(2 x \right)}

    State clearly what is being differentiated.

  2. 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.

  3. Recall the derivative of the natural logarithm

    ddx(ln⁡kx)=1x\frac{d}{dx}\left(\ln kx\right)=\frac{1}{x}

    The constant k cancels, so ln(kx) and ln(x) share the derivative 1/x.

  4. Differentiate the term e^{3 x}

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

    Differentiate this term using its standard result.

  5. Differentiate the term \ln{\left(2 x \right)}

    ddx(ln⁡(2x))=1x\frac{d}{dx}\left(\ln{\left(2 x \right)}\right)=\frac{1}{x}

    Differentiate this term using its standard result.

  6. Combine the differentiated terms

    dydx=3e3x+1x\frac{dy}{dx}=3 e^{3 x} + \frac{1}{x}

    Add the individual derivatives together.

  7. Recall that differentiation is linear

    ddx(f+g)=dfdx+dgdx\frac{d}{dx}\left(f+g\right)=\frac{df}{dx}+\frac{dg}{dx}

    Differentiate each term separately and add the results.

  8. Note that constant multiples are preserved

    ddx(c f)=c dfdx\frac{d}{dx}\left(c\,f\right)=c\,\frac{df}{dx}

    A constant factor multiplies straight through the derivative.

  9. Remember that constants differentiate to zero

    ddx(c)=0\frac{d}{dx}\left(c\right)=0

    Any constant term disappears when differentiated.

  10. Confirm every term has been differentiated

    all 2 term(s) processed\text{all }2\text{ term(s) processed}

    No term should be left unchanged in the gradient function.

  11. Check each result against the formula book

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

    Standard derivatives are provided in the formula booklet.

  12. Recall the derivative of a logarithm

    ddx(ln⁡kx)=1x\frac{d}{dx}\left(\ln kx\right)=\frac{1}{x}

    The constant inside a logarithm does not affect its derivative.

  13. Recall the derivatives of sine and cosine

    ddx(sin⁡kx)=kcos⁡kx, ddx(cos⁡kx)=−ksin⁡kx\frac{d}{dx}\left(\sin kx\right)=k\cos kx,\ \frac{d}{dx}\left(\cos kx\right)=-k\sin kx

    These standard trigonometric results come from the formula book.

  14. Compare the derivative with the options

    dydx=3e3x+1x\frac{dy}{dx}=3 e^{3 x} + \frac{1}{x}

    Choose the option equal to this expression.

  15. Select the correct option

    3e3x+1x3 e^{3 x} + \frac{1}{x}

    This matches the derivative found above.

Answer
3e3x+1x3 e^{3 x} + \frac{1}{x}

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