Differentiation: past paper questions

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• Question 5May/June 2026, paper 11Structured5 marks

  The equation of a curve is $$y=-\frac{1}{2}x+\sqrt{x^3-3x-2}\mathrm{.}$$ Find an equation of the normal to the curve at the point $(6,11)$.

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• Question 7Oct/Nov 2025, paper 11Structured8 marksHas a diagram

  The diagram shows a sector of a circle with centre $O$ and radius $r\text{ cm}$. The shaded region is bounded by the chord $AB$ and the arc $AB$. The size of angle $AOB$ is $\frac{2}{3}\pi \text{ radians}$.

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• Question 2Oct/Nov 2024, paper 11Structured4 marks

  The curve $y=x^2-\frac{a}{x}$ has a stationary point at $(-3,b)$. Find the values of the constants $a$ and $b$.

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• Question 3Oct/Nov 2023, paper 11Structured3 marksHas a diagram

  The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, $x\text{ cm}$, of each edge of the container increases at the constant rate of $0.01\text{ cm per minute}$.

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• Question 10May/June 2026, paper 11Structured11 marksHas a diagram

  The equation of a curve is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=3x^2-6$. The point $P(2,3)$ is a stationary point on the curve. By evaluating $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ at $P$, determine the nature of this stationary point.

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• Question 11Oct/Nov 2025, paper 11Structured11 marks

  A curve passes through the point $P(4,3)$ and is such that $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{8}{x^2}-\frac{10}{(2x-3{)}^2}\mathrm{.}$$ Find the equation of the normal to the curve at $P$.

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• Question 5Oct/Nov 2024, paper 11Structured7 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=4x-3\sqrt{x}+1$. Find the $x$-coordinate of the point on the curve at which the gradient is $\frac{11}{2}$.

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• Question 10Oct/Nov 2023, paper 11Structured11 marks

  A curve has a stationary point at $(2,-10)$ and is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=6x$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

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• Question 5May/June 2026, paper 12Structured7 marks

  The equation of a curve is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=6x^2+p$, where $p$ is a constant. The curve has minimum points at $(2,3)$ and $(-2,3)$. Find the coordinates of the other stationary point of the curve.

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• Question 4Oct/Nov 2025, paper 12Structured6 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=k{x}^3+\frac{2}{x^2}$, where $k$ is a constant. The curve passes through the point $S(2,20)$ and the gradient of the curve at $S$ is $\frac{65}{2}$. Find the value of $k$.

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• Question 7Oct/Nov 2024, paper 11Structured8 marks

  The diagram shows part of the curve with equation $y=\frac{12}{\sqrt[3]{2x+1}}$. The point $A$ on the curve has coordinates $(\frac{7}{2},6)$. Find the equation of the tangent to the curve at $A$.

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• Question 3Oct/Nov 2023, paper 12Structured6 marksHas a diagram

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2}x+\frac{72}{x^4}$. The curve passes through the point $P(2,8)$. Find the equation of the normal to the curve at $P$.

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• Question 9May/June 2026, paper 12Structured11 marksHas a diagram

  The diagram shows the curve with equation $y=3\sqrt{2x+1}$, the normal to the curve at the point $R(4,9)$ and the line $y=a$, where $a$ is a positive constant. Find the area of the shaded region.

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• Question 5Oct/Nov 2025, paper 12Structured8 marksHas a diagram

  The equation of a curve is $y=4x^{\frac{1}{2}}-x$. The curve has a maximum point when $x=a$ and crosses the $x$-axis at the point with coordinates $(b,0)$, where $b>0$. The shaded region is bounded by the curve, the line $x=a$ and the $x$-axis (see diagram).

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• Question 9Oct/Nov 2024, paper 11Structured8 marks

  The equation of a curve is $y=4+5x+6x^2-3x^3$. Find the set of values of $x$ for which $y$ decreases as $x$ increases.

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• Question 10Oct/Nov 2023, paper 12Structured7 marks

  The equation of a curve is $y=\mathrm{f}(x)$, where $\mathrm{f}(x)=(4x-3{)}^{\frac{5}{3}}-\frac{20}{3}x$. Find the $x$-coordinates of the stationary points of the curve and determine their nature.

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• Question 9Oct/Nov 2025, paper 12Structured8 marks

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{4}{(3x-6{)}^2}+\frac{1}{(3x-6{)}^3}$ for $x>2$. Find an expression for ${\mathrm{f}}^{\mathrm{\prime }}(x)$ and hence determine whether $\mathrm{f}$ is an increasing function, a decreasing function or neither.

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• Question 3Oct/Nov 2024, paper 12Structured5 marks

  The equation of a curve is $y=2x^2-3$. Two points $A$ and $B$ with $x$-coordinates 2 and $(2+h)$ respectively lie on the curve. Find and simplify an expression for the gradient of the chord $AB$ in terms of $h$.

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• Question 9Oct/Nov 2023, paper 13Structured8 marks

  A curve has equation $y=2x^{\frac{1}{2}}-1$. Find the equation of the normal to the curve at the point $A(4,3)$, giving your answer in the form $y=mx+c$.

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• Question 6May/June 2026, paper 13Structured7 marks

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{16}{x^3}+3x+5$ for $x\mathrm{\ne }0$. Find the coordinates of all stationary points on the curve $y=\mathrm{f}(x)$.

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• Question 3Oct/Nov 2025, paper 13Structured3 marks

  The equation of a curve is $y=\mathrm{f}(x)$, where $\mathrm{f}(x)=\frac{1}{2}{x}^{\frac{2}{3}}(x-2{)}^2$. The following points lie on the curve. Non-exact values of the $y$-coordinates are given correct to 6 decimal places.

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• Question 10Oct/Nov 2024, paper 12Structured10 marks

  A function $\mathrm{f}$ with domain $x>0$ is such that ${\mathrm{f}}^{\mathrm{\prime }}(x)=8(2x-3{)}^{\frac{1}{3}}-10x^{\frac{2}{3}}$. It is given that the curve with equation $y=\mathrm{f}(x)$ passes through the point $(1,0)$.

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• Question 11Oct/Nov 2023, paper 13Structured10 marksHas a diagram

  The diagram shows part of the curve with equation $y=x+\frac{2}{(2x-1{)}^2}$. The lines $x=1$ and $x=2$ intersect the curve at $P$ and $Q$ respectively and $R$ is the stationary point on the curve.

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• Question 9May/June 2026, paper 13Structured7 marks

  The equation of a curve is $y=\sqrt{3x+1}$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

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• Question 11Oct/Nov 2025, paper 13Structured13 marks

  The equation of a curve is $y=\frac{8}{3x-8}-\frac{6}{x-1}$. Find the coordinates of the point at which the tangent to the curve at the point $(3,5)$ intersects the line $y=-8x$.

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• Question 11Oct/Nov 2024, paper 13Structured12 marks

  The equation of a curve is $y=k{x}^{\frac{1}{2}}-4x^2+2$, where $k$ is a constant. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ in terms of $k$.

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• Question 9May/June 2023, paper 11Structured6 marks

  Water is poured into a tank at a constant rate of $500{\text{ cm}}^3$ per second. The depth of water in the tank, $t$ seconds after filling starts, is $h\text{ cm}$.

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• Question 2May/June 2025, paper 11Structured6 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=4(2x-5{)}^3-9x^{\frac{1}{2}}$. The curve passes through the point $A(4,-\frac{11}{2})$. Find the gradient of the normal to the curve at the point $A$.

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• Question 4May/June 2024, paper 11Structured3 marks

  The equation of a curve is $y=\mathrm{f}(x)$, where $\mathrm{f}(x)=(2x-1)\sqrt{3x-2}-2$. The following points lie on the curve. Non-exact values have been given correct to 5 decimal places.

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• Question 10May/June 2023, paper 11Structured11 marksHas a diagram

  The diagram shows part of the curve with equation $y=\frac{4}{(2x-1{)}^2}$ and parts of the lines $x=1$ and $y=1$. The curve passes through the points $A(1,4)$ and $B(\frac{3}{2},1)$.

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