differentiation, limits and applications

Sub-topic 1 2025 · National One-Eighth Free

Least perimeter of a rectangle of fixed area $96\ \text{cm}^2$: $p=2x+\dfrac{192}{x}$

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Sub-topic 2 2026 · National Preliminary Premium

Least perimeter of a rectangle of constant area $a$: $p_{\min}=4\sqrt{a}$

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Sub-topic 3 2025 · National One-Eighth Premium

$y=x^3-3x^2+8$ at $x=2$: increasing, stationary or a maximum? (true/false)

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Sub-topic 4 2024 · National One-Eighth Premium

Functions with no finite stationary point: $x$, $\dfrac{1}{x}$, $(x-2)^2$ (true/false)

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Sub-topic 5 2024 · National Preliminary Premium

Gradient of the tangent to $y^2=4(x+2)$ at $(2,4)$ by implicit differentiation

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Sub-topic 6 2024 · National Preliminary Premium

Least value of $2x^2-20x+30$ using differentiation

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Sub-topic 7 2026 · National One-Eighth Premium

Normal to $y=x^2-x+5$ parallel to $x+3y=10$

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Sub-topic 8 2025 · National Final Premium

Normal to $y=x^2-2x+2$ parallel to $x+2y=10$

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Sub-topic 9 2026 · National Preliminary Premium

$x$-values of the stationary points of $y=x^3-3x^2-45x+5$

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Sub-topic 10 2022 · National Final Premium

Stationary point of $y=x^3-3x^2+3x-2$

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Sub-topic 11 2026 · National Preliminary Premium

Stationary point of $y=x^2+\dfrac{16}{x}$

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Sub-topic 12 2025 · National Preliminary Premium

Gradient of $y=x^2-\dfrac{1}{x^2}$ at $x=1$

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Sub-topic 13 2026 · National Semifinal Premium

$\dfrac{dy}{dx}$ of $x^2-3xy+2y^2=6$ at $(1,-1)$ by implicit differentiation

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Sub-topic 14 2025 · National Semifinal Premium

Finding $b$, $c$, $d$ in $f(x)=3x^3+bx^2+cx+d$ from $f'(x)$ and $f(1)=8$

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Sub-topic 15 2026 · National Quarterfinal Premium

Points of inflection of $y=x^4-6x^2+9$

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Sub-topic 16 2026 · National One-Eighth Premium

Stationary point of $y=\sqrt{x^2+25}$

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Sub-topic 17 2026 · National One-Eighth Premium

Point of inflection of $y=x^3-6x^2+5x+8$

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Sub-topic 18 2024 · National Preliminary Premium

Point of inflection of $y=x^3-6x^2+5x+7$

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Sub-topic 19 2025 · National One-Eighth Premium

Point of inflection of $y=x^3-2x+5$

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Sub-topic 20 2018 · National Preliminary Premium

Differentiating $y=(x+1)(x-1)$ by expanding first

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Sub-topic 21 2018 · National One-Eighth Premium

Quotient rule: $\dfrac{2x-3}{x+1}$, $\dfrac{x+2}{2x-1}$, $\dfrac{x}{x+3}$

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Sub-topic 22 2026 · National Preliminary Premium

Quotient rule: $y=\dfrac{2x+1}{x-2}$

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Sub-topic 23 2025 · National Quarterfinal Premium

Stationary point of $y=\dfrac{(x+1)^2}{x^2}$

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Sub-topic 24 2026 · National Final Premium

Stationary points of $y=\dfrac{x^2}{x-1}$

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Sub-topic 25 2024 · National Final Premium

Stationary point of $y=\dfrac{(x+1)^2}{x^2}$

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Sub-topic 26 2023 · National One-Eighth Premium

Stationary points of $y=\sqrt{x^3-3x^2+8}$

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Sub-topic 27 2024 · National One-Eighth Premium

Stationary points of $y=x^3-3x^2+8$

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Sub-topic 28 2024 · National One-Eighth Premium

Points on a cubic curve with a given gradient (e.g. $y=x^3+6x^2+3x+3$, gradient $-6$)

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Sub-topic 29 2024 · National One-Eighth Premium

Point of inflection of $y=x^3+6x^2-6x-8$

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Sub-topic 30 2024 · Regional Championship Premium

Chain rule: $y=(3x^2-6x+5)^8$

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Sub-topic 31 2026 · National Preliminary Premium

$y=x^3-3x^2+8$ at $x=2$: increasing or stationary? (true/false)

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Sub-topic 32 2025 · National One-Eighth Premium

$y=x^3-3x^2+8$ at $x=2$: increasing or stationary? (true/false)

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Sub-topic 33 2023 · National Preliminary Premium

Limits of the form $\dfrac{0}{0}$: $\lim_{x\to a}\dfrac{x^2+bx+c}{x-a}$

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Sub-topic 34 2023 · National Preliminary Premium

$\lim_{x\to-4}\dfrac{x^2+2x-8}{x+4}$

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Sub-topic 35 2023 · National Preliminary Premium

Tangent to $y=2x^2-7x+9$ at $x=2$

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Sub-topic 36 2003 · National Semifinal Premium

Differentiating $y=x^5+6x^4+10x^2+8$ term by term

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Sub-topic 37 2003 · National Semifinal Premium

Differentiating $y=(3+x)(4-x)$

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Sub-topic 38 2012 · National Preliminary Premium

Differentiating $y=5x^3-4x^2+3x-10$ term by term

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Sub-topic 39 2012 · National Preliminary Premium

Differentiating $y=3x^2-\dfrac{4}{x^2}$

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Sub-topic 40 2013 · National Preliminary Premium

Minimum value of $y=\dfrac{x^2+16}{x}$

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Sub-topic 41 2013 · National Preliminary Premium

Differentiating $y=5x^4-6x^2-10$

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Sub-topic 42 2013 · National Preliminary Premium

Differentiating $y=3x^{-3}+5x^2-7x+8$

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Sub-topic 43 2013 · National Preliminary Premium

Differentiating $y=5x^3-\dfrac{6}{x^3}$

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Sub-topic 44 2013 · National Quarterfinal Premium

Product rule: $y=x^2\sin(2x)$

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Sub-topic 45 2013 · National Quarterfinal Premium

Quotient rule: $y=\dfrac{\sin x}{x^2}$

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Sub-topic 46 2013 · National Quarterfinal Premium

Chain rule: $y=(x+\sin x)^4$

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Sub-topic 47 2014 · National Final Premium

Rate of change of the volume of a sphere, $\dfrac{dv}{dt}$ at $r=10$ cm

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Sub-topic 48 2014 · National Final Premium

Rate of change of the surface area of a sphere, $\dfrac{da}{dt}$ at $r=10$ cm

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Sub-topic 49 2017 · National Quarterfinal Premium

$\dfrac{dy}{dx}$ of $x^2+2xy+3y^2=10$ (implicit differentiation)

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Sub-topic 50 2017 · National Quarterfinal Premium

$\dfrac{dy}{dx}$ of $2x^2+xy+y^2=15$ (implicit differentiation)

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Sub-topic 51 2017 · National Quarterfinal Premium

$\dfrac{dy}{dx}$ of $x^2-xy-y^2=20$ (implicit differentiation)

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Sub-topic 52 2021 · National Quarterfinal Premium

Turning points of $y=2x^3-3x^2+5$

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Sub-topic 53 2021 · National Quarterfinal Premium

Second derivative of $y=2x^4-3x^3+15x$

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Sub-topic 54 2021 · National Preliminary Premium

Chain rule: $y=(x^2+2x)^5$

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Sub-topic 55 2021 · National Semifinal Premium

No stationary point: $y=\dfrac{2x+1}{x-2}$

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Sub-topic 56 2023 · National Preliminary Premium

Stationary point of $y=\sqrt{x^2+4}$

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Sub-topic 57 2023 · National Quarterfinal Premium

No stationary point: $y=\dfrac{4x-3}{3x-4}$

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Sub-topic 58 2023 · National Quarterfinal Premium

Stationary points of $y=\sqrt{x^3-3x^2+8}$

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Sub-topic 59 2023 · Regional Championship Premium

Point of contact of the tangent $y=2x-1$ to $y=x^2-2x+3$

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Differentiation, limits and applications

Least perimeter of a rectangle of fixed area $96\ \text{cm}^2$: $p=2x+\dfrac{192}{x}$

Least perimeter of a rectangle of constant area $a$: $p_{\min}=4\sqrt{a}$

$y=x^3-3x^2+8$ at $x=2$: increasing, stationary or a maximum? (true/false)

Functions with no finite stationary point: $x$, $\dfrac{1}{x}$, $(x-2)^2$ (true/false)

Gradient of the tangent to $y^2=4(x+2)$ at $(2,4)$ by implicit differentiation

Least value of $2x^2-20x+30$ using differentiation

Normal to $y=x^2-x+5$ parallel to $x+3y=10$

Normal to $y=x^2-2x+2$ parallel to $x+2y=10$

$x$-values of the stationary points of $y=x^3-3x^2-45x+5$

Stationary point of $y=x^3-3x^2+3x-2$

Stationary point of $y=x^2+\dfrac{16}{x}$

Gradient of $y=x^2-\dfrac{1}{x^2}$ at $x=1$

$\dfrac{dy}{dx}$ of $x^2-3xy+2y^2=6$ at $(1,-1)$ by implicit differentiation

Finding $b$, $c$, $d$ in $f(x)=3x^3+bx^2+cx+d$ from $f'(x)$ and $f(1)=8$

Points of inflection of $y=x^4-6x^2+9$

Stationary point of $y=\sqrt{x^2+25}$

Point of inflection of $y=x^3-6x^2+5x+8$

Point of inflection of $y=x^3-6x^2+5x+7$

Point of inflection of $y=x^3-2x+5$

Differentiating $y=(x+1)(x-1)$ by expanding first

Quotient rule: $\dfrac{2x-3}{x+1}$, $\dfrac{x+2}{2x-1}$, $\dfrac{x}{x+3}$

Quotient rule: $y=\dfrac{2x+1}{x-2}$

Stationary point of $y=\dfrac{(x+1)^2}{x^2}$

Stationary points of $y=\dfrac{x^2}{x-1}$

Stationary point of $y=\dfrac{(x+1)^2}{x^2}$

Stationary points of $y=\sqrt{x^3-3x^2+8}$

Stationary points of $y=x^3-3x^2+8$

Points on a cubic curve with a given gradient (e.g. $y=x^3+6x^2+3x+3$, gradient $-6$)

Point of inflection of $y=x^3+6x^2-6x-8$

Chain rule: $y=(3x^2-6x+5)^8$

$y=x^3-3x^2+8$ at $x=2$: increasing or stationary? (true/false)

$y=x^3-3x^2+8$ at $x=2$: increasing or stationary? (true/false)

Limits of the form $\dfrac{0}{0}$: $\lim_{x\to a}\dfrac{x^2+bx+c}{x-a}$

$\lim_{x\to-4}\dfrac{x^2+2x-8}{x+4}$

Tangent to $y=2x^2-7x+9$ at $x=2$

Differentiating $y=x^5+6x^4+10x^2+8$ term by term

Differentiating $y=(3+x)(4-x)$

Differentiating $y=5x^3-4x^2+3x-10$ term by term

Differentiating $y=3x^2-\dfrac{4}{x^2}$

Minimum value of $y=\dfrac{x^2+16}{x}$

Differentiating $y=5x^4-6x^2-10$

Differentiating $y=3x^{-3}+5x^2-7x+8$

Differentiating $y=5x^3-\dfrac{6}{x^3}$

Product rule: $y=x^2\sin(2x)$

Quotient rule: $y=\dfrac{\sin x}{x^2}$

Chain rule: $y=(x+\sin x)^4$

Rate of change of the volume of a sphere, $\dfrac{dv}{dt}$ at $r=10$ cm

Rate of change of the surface area of a sphere, $\dfrac{da}{dt}$ at $r=10$ cm

$\dfrac{dy}{dx}$ of $x^2+2xy+3y^2=10$ (implicit differentiation)

$\dfrac{dy}{dx}$ of $2x^2+xy+y^2=15$ (implicit differentiation)

$\dfrac{dy}{dx}$ of $x^2-xy-y^2=20$ (implicit differentiation)

Turning points of $y=2x^3-3x^2+5$

Second derivative of $y=2x^4-3x^3+15x$

Chain rule: $y=(x^2+2x)^5$

No stationary point: $y=\dfrac{2x+1}{x-2}$

Stationary point of $y=\sqrt{x^2+4}$

No stationary point: $y=\dfrac{4x-3}{3x-4}$

Stationary points of $y=\sqrt{x^3-3x^2+8}$

Point of contact of the tangent $y=2x-1$ to $y=x^2-2x+3$