Hard A-Level Standard derivatives Questions

Challenging, exam-style A-Level Standard derivatives questions with worked solutions. Stretch yourself on the hardest differentiation, standard derivatives problems.

differentiationstandard derivatives
A-Level34 questionsStep-by-step solutions
Question 1
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}
Question 2
8 markschallenging
Which expression is the derivative of f(x)=3x2+4sin⁡(2x)f(x)=3 x^{2} + 4 \sin{\left(2 x \right)} with respect to xx?
Show worked solution

Worked solution

  1. Write down the function

    y=3x2+4sin⁡(2x)y=3 x^{2} + 4 \sin{\left(2 x \right)}

    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. Recall the derivative of sine

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

    Differentiating sine gives cosine, times the constant k.

  4. Differentiate the term 3 x^{2}

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

    Differentiate this term using its standard result.

  5. Differentiate the term 4 \sin{\left(2 x \right)}

    ddx(4sin⁡(2x))=8cos⁡(2x)\frac{d}{dx}\left(4 \sin{\left(2 x \right)}\right)=8 \cos{\left(2 x \right)}

    Differentiate this term using its standard result.

  6. Combine the differentiated terms

    dydx=6x+8cos⁡(2x)\frac{dy}{dx}=6 x + 8 \cos{\left(2 x \right)}

    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=6x+8cos⁡(2x)\frac{dy}{dx}=6 x + 8 \cos{\left(2 x \right)}

    Choose the option equal to this expression.

  15. Select the correct option

    6x+8cos⁡(2x)6 x + 8 \cos{\left(2 x \right)}

    This matches the derivative found above.

Answer
6x+8cos⁡(2x)6 x + 8 \cos{\left(2 x \right)}
Question 3
8 markschallenging
Which of the following is dydx\frac{dy}{dx} for y=x3−3x+ex+cos⁡(x)y=x^{3} - 3 x + e^{x} + \cos{\left(x \right)}?
Show worked solution

Worked solution

  1. Write down the function

    y=x3−3x+ex+cos⁡(x)y=x^{3} - 3 x + e^{x} + \cos{\left(x \right)}

    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. 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.

  4. Recall the derivative of cosine

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

    Differentiating cosine gives minus sine, times the constant k.

  5. Differentiate the term x^{3}

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

    Differentiate this term using its standard result.

  6. Differentiate the term - 3 x

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

    Differentiate this term using its standard result.

  7. Differentiate the term e^{x}

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

    Differentiate this term using its standard result.

  8. Differentiate the term \cos{\left(x \right)}

    ddx(cos⁡(x))=−sin⁡(x)\frac{d}{dx}\left(\cos{\left(x \right)}\right)=- \sin{\left(x \right)}

    Differentiate this term using its standard result.

  9. Combine the differentiated terms

    dydx=3x2+ex−sin⁡(x)−3\frac{dy}{dx}=3 x^{2} + e^{x} - \sin{\left(x \right)} - 3

    Add the individual derivatives together.

  10. 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.

  11. 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.

  12. Remember that constants differentiate to zero

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

    Any constant term disappears when differentiated.

  13. Confirm every term has been differentiated

    all 4 term(s) processed\text{all }4\text{ term(s) processed}

    No term should be left unchanged in the gradient function.

  14. Compare the derivative with the options

    dydx=3x2+ex−sin⁡(x)−3\frac{dy}{dx}=3 x^{2} + e^{x} - \sin{\left(x \right)} - 3

    Choose the option equal to this expression.

  15. Select the correct option

    3x2+ex−sin⁡(x)−33 x^{2} + e^{x} - \sin{\left(x \right)} - 3

    This matches the derivative found above.

Answer
3x2+ex−sin⁡(x)−33 x^{2} + e^{x} - \sin{\left(x \right)} - 3
Question 4
8 markschallenging
Which of the following is dydx\frac{dy}{dx} for y=x2+sin⁡(4x)+e−2xy=x^{2} + \sin{\left(4 x \right)} + e^{- 2 x}?
Show worked solution

Worked solution

  1. Write down the function

    y=x2+sin⁡(4x)+e−2xy=x^{2} + \sin{\left(4 x \right)} + e^{- 2 x}

    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. Recall the derivative of sine

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

    Differentiating sine gives cosine, times the constant k.

  4. 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.

  5. Differentiate the term x^{2}

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

    Differentiate this term using its standard result.

  6. Differentiate the term \sin{\left(4 x \right)}

    ddx(sin⁡(4x))=4cos⁡(4x)\frac{d}{dx}\left(\sin{\left(4 x \right)}\right)=4 \cos{\left(4 x \right)}

    Differentiate this term using its standard result.

  7. Differentiate the term e^{- 2 x}

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

    Differentiate this term using its standard result.

  8. Combine the differentiated terms

    dydx=2x+4cos⁡(4x)−2e−2x\frac{dy}{dx}=2 x + 4 \cos{\left(4 x \right)} - 2 e^{- 2 x}

    Add the individual derivatives together.

  9. 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.

  10. 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.

  11. Remember that constants differentiate to zero

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

    Any constant term disappears when differentiated.

  12. Confirm every term has been differentiated

    all 3 term(s) processed\text{all }3\text{ term(s) processed}

    No term should be left unchanged in the gradient function.

  13. 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.

  14. Compare the derivative with the options

    dydx=2x+4cos⁡(4x)−2e−2x\frac{dy}{dx}=2 x + 4 \cos{\left(4 x \right)} - 2 e^{- 2 x}

    Choose the option equal to this expression.

  15. Select the correct option

    2x+4cos⁡(4x)−2e−2x2 x + 4 \cos{\left(4 x \right)} - 2 e^{- 2 x}

    This matches the derivative found above.

Answer
2x+4cos⁡(4x)−2e−2x2 x + 4 \cos{\left(4 x \right)} - 2 e^{- 2 x}
Question 5
8 markschallenging
Which of the following is dydx\frac{dy}{dx} for y=2x4+ln⁡(x)−cos⁡(3x)y=2 x^{4} + \ln{\left(x \right)} - \cos{\left(3 x \right)}?
Show worked solution

Worked solution

  1. Write down the function

    y=2x4+ln⁡(x)−cos⁡(3x)y=2 x^{4} + \ln{\left(x \right)} - \cos{\left(3 x \right)}

    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. 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. Recall the derivative of cosine

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

    Differentiating cosine gives minus sine, times the constant k.

  5. Differentiate the term 2 x^{4}

    ddx(2x4)=8x3\frac{d}{dx}\left(2 x^{4}\right)=8 x^{3}

    Differentiate this term using its standard result.

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

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

    Differentiate this term using its standard result.

  7. Differentiate the term - \cos{\left(3 x \right)}

    ddx(−cos⁡(3x))=3sin⁡(3x)\frac{d}{dx}\left(- \cos{\left(3 x \right)}\right)=3 \sin{\left(3 x \right)}

    Differentiate this term using its standard result.

  8. Combine the differentiated terms

    dydx=8x3+3sin⁡(3x)+1x\frac{dy}{dx}=8 x^{3} + 3 \sin{\left(3 x \right)} + \frac{1}{x}

    Add the individual derivatives together.

  9. 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.

  10. 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.

  11. Remember that constants differentiate to zero

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

    Any constant term disappears when differentiated.

  12. Confirm every term has been differentiated

    all 3 term(s) processed\text{all }3\text{ term(s) processed}

    No term should be left unchanged in the gradient function.

  13. 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.

  14. Compare the derivative with the options

    dydx=8x3+3sin⁡(3x)+1x\frac{dy}{dx}=8 x^{3} + 3 \sin{\left(3 x \right)} + \frac{1}{x}

    Choose the option equal to this expression.

  15. Select the correct option

    8x3+3sin⁡(3x)+1x8 x^{3} + 3 \sin{\left(3 x \right)} + \frac{1}{x}

    This matches the derivative found above.

Answer
8x3+3sin⁡(3x)+1x8 x^{3} + 3 \sin{\left(3 x \right)} + \frac{1}{x}

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