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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>It is given that&nbsp;$y = \sin^{-1} px$, where&nbsp;$p$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph"><p>By further differentiation of the result in part <strong>(a)</strong> find, in terms of $p$, the Maclaurin series for&nbsp;$y$ up to and including the term in $x^3$.</p><span class="inline-marks">[5]</span></div></li><li data-list-item-id="imported-item-block-1-item-2" data-list-item-path="imported-item-block-1-item-2"><div class="list-item-paragraph"><p>Use your answer to part <strong>(b)</strong> to find the series expansion of $\frac{1}{\sqrt{1 - 9x^2}}$, up to and including the term in $x^2$.</p><span class="inline-marks">[3]</span></div></li></ol>

It is given that $y = \sin^{-1} px$, where $p$ is a constant.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y$.

Maclaurin Series from Differential Equations

Derivatives of Inverse Trigonometric Functions

Derivatives of Reciprocal Trigonometric Functions

Maclaurin Series by Successive Differentiation

Higher Derivatives by Implicit Differentiation

Problems involving Calculus

Deriving Series by Differentiation and Integration

N2025 P1 Q9

Techniques of Differentiation

Maclaurin Series and Power Series

<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.

Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.

Integration by the Reverse Chain Rule

Integration as an Inverse Process of Differentiation

Chain Rule

2016 DHS C1BT Q2

Techniques of Integration

<p>Differentiate each of the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\cos^{-1}\left(3x^2\right)$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$\ln\left(\frac{\sqrt{2x+1}}{x^3}\right)$.<span class="inline-marks">[3]</span></div></li></ol>

Differentiate each of the following expressions with respect to $x$.

Derivatives of Natural Logarithmic Functions

2024 JPJC Promo Q1

<p>The point $P$ travels along the curve $C$ with equation $y = x \sin^{-1} x$, $-1 &lt; x &lt; 1$. Let the gradient of the curve $C$ at the point $P$ be $m$.</p>
<p>If the $x$-coordinate of $P$ is increasing at the rate of $9$ units per second when&nbsp;$x = \frac{1}{2}$, find the exact value of the rate at which $m$ is changing at this instant.<span class="inline-marks">[4]</span></p>

The point $P$ travels along the curve $C$ with equation $y = x \sin^{-1} x$, $-1 < x < 1$. Let the gradient of the curve $C$ at the point $P$ be $m$.

If the $x$-coordinate of $P$ is increasing at the rate of $9$ units per second when $x = \frac{1}{2}$, find the exact value of the rate at which $m$ is changing at this instant.

Connected Rates of Change

Product Rule

Second and Higher Derivatives

2025 RI P1 Q2

Applications of Differentiation

<p>It is given that&nbsp;$y = \mathrm{e}^{2\sin^{-1} x}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that&nbsp;$\sqrt{1 - x^2} \frac{\mathrm{d}y}{\mathrm{d}x} = 2y$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph"><p>By further differentiation of the result in part <strong>(i)</strong>, find the Maclaurin series for&nbsp;$y$ up to and including the term in $x^2$.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using your result from part <strong>(ii)</strong>, find an approximate value for&nbsp;$\displaystyle \displaystyle\int_{0}^{0.1} x\mathrm{e}^{2\sin^{-1} x} \, \mathrm{d}x$, giving your answer to $4$ significant figures.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \mathrm{e}^{2\sin^{-1} x}$.

Show that $\sqrt{1 - x^2} \frac{\mathrm{d}y}{\mathrm{d}x} = 2y$.

Approximations Using Power Series

Derivatives of Exponential Functions

2025 MI P1 Q8

<p>It is given that $y={{e}^{{{\tan }^{-1}}4x}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $\left( 1+16{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+k\frac{\mathrm{d}y}{\mathrm{d}x}\left( 8x-1 \right)=0$, where $k$ is a constant to be found.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)</strong>, find the Maclaurin series for $y$ up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The first three non-zero terms of the Maclaurin expansion of ${{e}^{{{\tan }^{-1}}4x}}$ are equal to the first three non-zero terms of the series expansion of $\ln \left( \frac{a}{1-bx} \right)$, where $a$ and $b$ are constants. Using standard series from the List of Formulae (MF27), find the exact values of $a$ and $b$.</div><p style="text-align: right;">[2]</p></li></ol>

Show that $(1+16x^2)\frac{d^2y}{dx^2}+k\frac{dy}{dx}(8x-1)=0$, where $k$ is a constant to be found.

By further differentiation of the result in part (a), find the Maclaurin series for $y$ up to and including the term in $x^3$.

The first three non-zero terms of the Maclaurin expansion of $e^{\tan^{-1}4x}$ are equal to the first three non-zero terms of the series expansion of $\ln\left(\frac{a}{1-bx}\right)$, where $a$ and $b$ are constants. Using standard series from the List of Formulae (MF27), find the exact values of $a$ and $b$.

Finding Parameters from Power Series Coefficients

Maclaurin Series Using Standard Series

2025 DHS Promo Q6

<p>Given that&nbsp;$y = e^{\tan^{-1}x}$, show that&nbsp;$(1 + x^{2})\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} + (2x - 1)\frac{\mathrm{d}y}{\mathrm{d}x} = 0$.</p>

Given that $y = e^{\tan^{-1}x}$, show that $(1 + x^{2})\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} + (2x - 1)\frac{\mathrm{d}y}{\mathrm{d}x} = 0$.

<p>Given that&nbsp;$y={{({{\sin }^{-1}}x)}^{2}}$, show that&nbsp;$(1-{{x}^{2}}){{\left(\frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=4y$ and $(1-{{x}^{2}})\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2$.</p>

Given that $y={{({{\sin }^{-1}}x)}^{2}}$, show that $(1-{{x}^{2}}){{\left(\frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=4y$ and $(1-{{x}^{2}})\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2$.

<p>Find the $\frac{\mathrm{d}y}{\mathrm{d}x}$ of the following equations in terms of $x$ and $y$:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$x^{3} + y^{2} + xy = \sin y$</div></li><li><div class="list-item-paragraph">$y^{3} - 2xy = \sin y$</div></li><li><div class="list-item-paragraph">$y = x^{x}$</div></li><li><div class="list-item-paragraph">$x^{2} + 4y^{2} = 3yx$</div></li><li><div class="list-item-paragraph">$5x^{3} - 3y^{3} = 3yx$</div></li><li><div class="list-item-paragraph">$\tan y + x^{2} = x\ln y$</div></li><li><div class="list-item-paragraph">$\cos^{-1}(e^{y}) = x + 2y$</div></li></ol>

Find the $\frac{\mathrm{d}y}{\mathrm{d}x}$ of the following equations in terms of $x$ and $y$:

Implicit Differentiation

Logarithmic Differentiation

<p>Find the derivative of the following functions with respect to $x$:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$y = \sin^{-1}(\ln x)$</div></li><li><div class="list-item-paragraph">$y = \cos^{-1}(3x - 1)$</div></li><li><div class="list-item-paragraph">$y = \tan^{-1}(3^{x})$</div></li></ol>

Find the derivative of the following functions with respect to $x$:

Derivatives of Inverse Trigonometric functions (Using MF 27) Q1

<p>Given that $y={{\sin }^{-1}}x$, prove that $\left( 1-{{x}^{2}} \right)\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}-3x\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-\frac{\mathrm{d}y}{\mathrm{d}x}=0$.</p>

Given that $y={{\sin }^{-1}}x$, prove that $\left( 1-{{x}^{2}} \right)\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}-3x\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-\frac{\mathrm{d}y}{\mathrm{d}x}=0$.

Practice 4 Q20

<ol class="i-list"><li><div class="list-item-paragraph">Express $\frac{\mathrm{d}}{\mathrm{d}x}\left[ {{\tan }^{-1}}(xy) \right]$ in terms of $x$, $y$ and $\frac{\mathrm{d}y}{\mathrm{d}x}$.</div><div class="list-item-paragraph">It is given that $x$ and $y$ satisfy the equation:</div><div class="list-item-paragraph" style="text-align: center">${{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}(xy)=\frac{7}{12}\pi $.</div></li><li><div class="list-item-paragraph">Find the value of $y$ when $x=1$.</div></li><li><div class="list-item-paragraph">Show that, when $x=1$, $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{3}-\frac{1}{2\sqrt{3}}$.</div></li></ol>

Derivative as the Gradient of a Curve

Differentiation Involving Inverse Trigonometric Functions and Implicit Relations - AOD

<p>As part of a game, a person stands $x$ m away from a wall and tries to throw an object through a $0.40$ m wide hole in the wall.</p>
<p>The point $B$ denotes the bottom of the hole, which is located $3.05$ m above the ground. The object is thrown from point $P$ at a height of $1.75$ m above the ground. The angle $APB$ is denoted by $\theta $ in radians.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1771990853975-v2l71wiuygf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE5OTA4NTM5NzUtdjJsNzF3aXV5Z2YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTE5NCwiZXhwIjoxNzkyNzg5NDk0fQ.Lf5xHySFHRDoI4LWHvgEbP7cR5Xsc_CpyE7wf-i8Vxs" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-1761023681762-4unpjp7il" data-list-item-path="item-1761023681762-4unpjp7il"><div class="list-item-paragraph">Show that&nbsp;$\theta ={{\tan }^{-1}}\left( \frac{1.7}{x} \right)-{{\tan }^{-1}}\left( \frac{1.3}{x} \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761023682415-6fi0whjy1" data-list-item-path="item-1761023682415-6fi0whjy1"><div class="list-item-paragraph">Using differentiation, find the value of $x$ which gives the stationary value of $\theta $, giving your answer to four significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761023683713-hma2fmg80" data-list-item-path="item-1761023683713-hma2fmg80"><div class="list-item-paragraph">Show that the value of $x$ in <strong>(ii)</strong> corresponds to a maximum value of $\theta $, and suggest why knowing this value of $x$ may be useful in this situation.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761023690935-asp3laz1h" data-list-item-path="item-1761023690935-asp3laz1h"><div class="list-item-paragraph">The person walks away from the wall at a constant rate of $0.1$ m/s. Find the rate of change of $\theta $ at the instant when the person is $1$ m away from the wall.<span class="inline-marks">[2]</span></div></li></ol>

As part of a game, a person stands $x$ m away from a wall and tries to throw an object through a $0.40$ m wide hole in the wall.

The point $B$ denotes the bottom of the hole, which is located $3.05$ m above the ground. The object is thrown from point $P$ at a height of $1.75$ m above the ground. The angle $APB$ is denoted by $\theta $ in radians.

Problems Involving Maximisation and Minimisation

2024 ACJC Promo Q9

<p>Differentiate each of the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760848094536-lagej8vfz" data-list-item-path="item-1760848094536-lagej8vfz"><div class="list-item-paragraph">${{\tan }^{-1}}\sqrt{2+{{x}^{3}}}$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760848095458-ucu58ihkn" data-list-item-path="item-1760848095458-ucu58ihkn"><div class="list-item-paragraph">$\ln \left( \frac{x}{\sqrt{1-x}} \right)$<span class="inline-marks">[3]</span></div></li></ol>

Differentiate each of the following expressions with respect to $x$.

Product, Quotient and Chain Rules

2024 CJC Promo Q5

<p>It is given that&nbsp;$\mathrm{f}\left( x \right)={{\mathrm{e}}^{{{\tan }^{-1}}x}}$, where&nbsp;${{\tan }^{-1}}x$ denotes the principal values.</p>
<ol class="i-list"><li data-list-item-id="item-1760747941731-9vyv1phos" data-list-item-path="item-1760747941731-9vyv1phos"><div class="list-item-paragraph">Show that&nbsp;$\left( 1+{{x}^{2}} \right)\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)=\mathrm{f}\left( x \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760747942428-8ls2y4o9j" data-list-item-path="item-1760747942428-8ls2y4o9j"><div class="list-item-paragraph">By further differentiation of the above result, find the Maclaurin series for&nbsp;$\mathrm{f}(x)$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760747943185-0r00ysebb" data-list-item-path="item-1760747943185-0r00ysebb"><div class="list-item-paragraph">Using the series in part <strong>(ii)</strong> find an approximate value of $\displaystyle\displaystyle \displaystyle\int_{0}^{0.5}{\mathrm{f}\left( x \right)\,}\mathrm{d}x$, giving your answer to $4$ significant figures.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760747944566-nz3f4hhzo" data-list-item-path="item-1760747944566-nz3f4hhzo"><div class="list-item-paragraph">Comment on the suitability of substituting $x=1$ into the series in part <strong>(ii)</strong> to estimate the value of ${{\mathrm{e}}^{\frac{\pi }{4}}}$.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $\mathrm{f}\left( x \right)={{\mathrm{e}}^{{{\tan }^{-1}}x}}$, where ${{\tan }^{-1}}x$ denotes the principal values.

Series from Products, Substitution, Integration and Differentiation

Accuracy of Power Series Approximations

2024 JPJC P2 Q3

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Differentiate ${{\sin }^{-1}}\left( {{\mathrm{e}}^{2x}} \right)$ with respect to $x$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence find $\displaystyle\int{\frac{2{{\mathrm{e}}^{2x}}+{{\mathrm{e}}^{4x}}}{\sqrt{1-{{\mathrm{e}}^{4x}}}}}\,\mathrm{d}x$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that ${{\sin }^{n-2}}\theta {{\cos }^{2}}\theta ={{\sin }^{n-2}}\theta -{{\sin }^{n}}\theta $, where $n\in {{\mathbb{Z}}^{+}}$, $n\ge 2$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">By considering ${{\sin }^{n}}\theta ={{\sin }^{n-1}}\theta \sin \theta $, show that</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{n}}\theta \,\,\mathrm{d}\theta =\frac{n-1}{n}\displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{n-2}}\theta \,\,\mathrm{d}\theta }}$,</div><div class="list-item-paragraph">where $n\in {{\mathbb{Z}}^{+}}$, $n\ge 2$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence find the exact value of $\displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{3}}\theta \,\,\mathrm{d}\theta }$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Reduction Formulae for Integrals

Integration by Parts

Integrable Forms of Trigonometric Functions

2023 MI P1 Q10

Applications of Integration

<p>Differentiate each of the following expressions with respect to $x$.</p>
<ol class="i-list"><li data-list-item-id="item-1760448383610-65ijxbjmp" data-list-item-path="item-1760448383610-65ijxbjmp"><div class="list-item-paragraph">${{\tan }^{-1}}\sqrt{x+1}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760448384270-hdkvxskeh" data-list-item-path="item-1760448384270-hdkvxskeh"><div class="list-item-paragraph">$\ln \left( \frac{x+{{\mathrm{e}}^{x}}}{{{3}^{x}}} \right)$.<span class="inline-marks">[3]</span></div></li></ol>

2023 ACJC Promo Q1

<ol class="abc-list"><li data-list-item-id="item-1760448286868-2bmorn6dz" data-list-item-path="item-1760448286868-2bmorn6dz"><div class="list-item-paragraph">Find $\frac{\mathrm{d}}{\mathrm{d}x}x{{\tan }^{-1}}\left( 2x \right)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760448287400-0r2kda7qz" data-list-item-path="item-1760448287400-0r2kda7qz"><div class="list-item-paragraph">Find $\frac{\mathrm{d}}{\mathrm{d}x}\ln \left( \sqrt{\frac{{{x}^{2}}+1}{x}} \right)$, leaving your answer as a single fraction.<span class="inline-marks">[3]</span></div></li></ol>

2023 CJC Promo Q2

<p>It is given that&nbsp;$y={{\sin }^{-1}}x$, $-1\le x\le 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760430006158-aony6z2na" data-list-item-path="item-1760430006158-aony6z2na"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=x{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430006939-x6lgv1h1z" data-list-item-path="item-1760430006939-x6lgv1h1z"><div class="list-item-paragraph">Using the result in part <strong>(a)</strong>, find the Maclaurin expansion for&nbsp;$y$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430007540-7hyj4gagf" data-list-item-path="item-1760430007540-7hyj4gagf"><div class="list-item-paragraph">By substituting $x=0.5$ into the expansion found in part <strong>(b)</strong>, find an approximate value of $\pi $.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y={{\sin }^{-1}}x$, $-1\le x\le 1$.

2023 CJC Promo Q5

<ol class="i-list"><li><div class="list-item-paragraph">Given that $\ln \left( ky \right)={{\tan }^{-1}}\left( kx \right)$, where $k$ is a non-zero constant, show that $\left( 1+{{k}^{2}}{{x}^{2}} \right)\frac{\mathrm{d}y}{\mathrm{d}x}=ky$. By further differentiation of this result, find the Maclaurin’s series for $y$, in terms of $k$, up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Let $k=1$. By using a suitable substitution for $~x$, find, in terms of $\sqrt{3}$, an approximate value of ${{\mathrm{e}}^{\frac{\pi }{6}}}$.</div><p style="text-align: right;">[4]</p></li></ol>

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