Differentiation: past paper questions

Where the questions come from

• Question 8Oct/Nov 2025, paper 21Structured6 marks

  The equation of a curve is $y=4{\mathrm{e}}^{1-2x}\sqrt{3x-1}$. Find the exact coordinates of the stationary point of the curve.

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• Question 7Oct/Nov 2023, paper 21Structured11 marks

  The curve with equation ${\mathrm{e}}^{2x}-18x+y^3+y=11$ has a stationary point at $(p,q)$. Find the exact value of $p$.

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• Question 2Oct/Nov 2024, paper 22Structured6 marks

  Let $\mathrm{f}(x)=4{\sin }^{2}3x$. Find the value of ${\mathrm{f}}^{\mathrm{\prime }}\left(\frac{1}{4}\pi \right)$.

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• Question 3Oct/Nov 2023, paper 22Structured5 marksHas a diagram

  The diagram shows the curve with equation $y=6{\mathrm{e}}^{-\frac{1}{2}x}$. The points on the curve with $x$-coordinates 0 and 2 are denoted by $A$ and $B$ respectively.

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• Question 7Oct/Nov 2025, paper 22Structured8 marks

  A curve has equation $5x^{2}y+4{\mathrm{e}}^{2y}-7x+10=0$. Find an expression in terms of $x$ and $y$ for $\frac{\mathrm{d}y}{\mathrm{d}x}$ and hence find the gradient of the curve at the point for which $y=0$.

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• Question 3Oct/Nov 2024, paper 22Structured6 marks

  A curve has equation $6{\mathrm{e}}^{-x}{y}^2+{\mathrm{e}}^{2x}-12y+7=0$. Find the gradient of the curve at the point $(\ln 3,2)$.

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• Question 2Oct/Nov 2023, paper 23Structured5 marks

  A curve has equation $y=3\tan \frac{1}{2}x\cos 2x$. Find the gradient of the curve at the point for which $x=\frac{1}{3}\pi$.

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• Question 1May/June 2025, paper 21Structured2 marks

  Given that $y=6x\cos (x^2+1)$, find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

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• Question 4May/June 2025, paper 21Structured7 marksHas a diagram

  The diagram shows the curve with equation $y=6{\mathrm{e}}^{2x}-{\mathrm{e}}^{3x}$. The shaded region is bounded by the axes and the curve. Find the exact $x$-coordinate of the maximum point.

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• Question 2May/June 2024, paper 21Structured5 marks

  A curve has equation $x^2\ln y+y^2+4x=9$. Find the gradient of the curve at the point $(2,1)$.

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• Question 2May/June 2023, paper 21Structured5 marks

  A curve has equation $y=\frac{2+3\ln x}{1+2x}$. Find the equation of the tangent to the curve at the point $(1,\frac{2}{3})$.

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• Question 3May/June 2025, paper 22Structured5 marks

  Find the coordinates of the stationary points of the curve with equation $y=\frac{8x}{2x+3}-6x+5$.

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• Question 3May/June 2024, paper 22Structured8 marksHas a diagram

  The diagram shows the curve with equation $y=8{\mathrm{e}}^{-x}-{\mathrm{e}}^{2x}$. The curve crosses the $y$-axis at the point $A$ and the $x$-axis at the point $B$. The shaded region is bounded by the curve and the two axes.

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• Question 5May/June 2023, paper 22Structured9 marksHas a diagram

  The diagram shows the curve with equation $y={\mathrm{e}}^{-\frac{1}{2}x}(x^2-5x+4)$. The curve crosses the $x$-axis at the points $A$ and $B$, and has a maximum at the point $C$. Find the exact gradient of the curve at $B$.

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• Question 6May/June 2025, paper 22Structured9 marks

  A curve has equation $(x^2-3)\ln y+6x=14$. Show that there is no point on the curve at which the $y$-coordinate is ${\mathrm{e}}^{-1}$.

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• Question 6May/June 2024, paper 22Structured9 marksHas a diagram

  The diagram shows the curve with equation $y=\frac{\ln (2x+1)}{x+3}$. The curve has a maximum point $M$. Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

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