Differentiation: past paper questions

Where the questions come from

• Question 3May/June 2026, paper 31Structured6 marks

  The equation of a curve is $y=\frac{2x^2+7x}{3x^2-1}$. Find an equation of the normal to the curve at the point where $x=1$.

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• Question 3Oct/Nov 2024, paper 31Structured4 marks

  The equation of a curve is $\ln (x+y)=3x^{2}y$. Find the gradient of the curve at the point $(1,0)$.

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• Question 1Oct/Nov 2023, paper 31Structured5 marks

  Find the exact coordinates of the points on the curve $y=\frac{x^2}{1-3x}$ at which the gradient of the tangent is equal to 8.

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• Question 6May/June 2026, paper 31Structured10 marks

  A curve has the equation $y={\mathrm{e}}^{(x^2+4x)}{\cos }^{2}x$, where $\cos x\mathrm{\ne }0$. Show that the $x$-coordinates of the stationary points of this curve satisfy the equation $\tan x=x+2$.

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• Question 7Oct/Nov 2025, paper 32Structured8 marks

  The equation of a curve is $2y^3-3x^{2}y-x^3=16$. Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x^2+2xy}{2y^2-x^2}$.

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• Question 8Oct/Nov 2025, paper 33Structured10 marks

  The curve with equation $y={\mathrm{e}}^{-5x}\ln 5x$ has a stationary point at $x=p$. Show that $p$ satisfies the equation $\ln 5p=\frac{1}{5p}$.

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• Question 7Oct/Nov 2023, paper 33Structured8 marks

  The equation of a curve is $x^3+y^2+3x^2+3y=4$. Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{3x^2+6x}{2y+3}$.

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• Question 10May/June 2024, paper 31Structured10 marks

  Given that $2x=\tan y$, show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{1+4x^2}$.

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• Question 5May/June 2025, paper 33Structured6 marks

  The equation of a curve is $xy+y^2{\mathrm{e}}^{-x}=4$. Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y^2-y{\mathrm{e}}^x}{x{\mathrm{e}}^x+2y}$.

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• Question 4May/June 2024, paper 32Structured6 marks

  The equation of a curve is $y{\mathrm{e}}^{2x}+y^2{\mathrm{e}}^x=6$. Find the gradient of the curve at the point where $y=1$.

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• Question 5May/June 2023, paper 31Structured8 marks

  The equation of a curve is $x^{2}y-a{y}^2=4a^3$, where $a$ is a non-zero constant. Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2xy}{2ay-x^2}$.

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• Question 6May/June 2024, paper 32Structured9 marksHas a diagram

  The diagram shows the curve $y=x{\mathrm{e}}^{-ax}$, where $a$ is a positive constant, and its maximum point $M$. Find the exact coordinates of $M$.

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• Question 7May/June 2023, paper 32Structured9 marks

  The equation of a curve is $3x^2+4xy+3y^2=5$. Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{3x+2y}{2x+3y}$.

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• Question 10May/June 2023, paper 32Structured10 marksHas a diagram

  The diagram shows the curve $y=(x+5)\sqrt{3-2x}$ and its maximum point $M$. Find the exact coordinates of $M$.

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• Question 9May/June 2024, paper 33Structured11 marksHas a diagram

  A container in the shape of a cuboid has a square base of side $x$ and a height of $(10-x)$. It is given that $x$ varies with time, $t$, where $t>0$. The container decreases in volume at a rate which is inversely proportional to $t$.

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