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<p>Show that:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\sin^{-1}x = \frac{1}{\sqrt{1 - x^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}{{\cos }^{-1}}x=-\frac{1}{\sqrt{1-{{x}^{2}}}}$</div></li></ol>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}x = \frac{1}{1 + x^{2}}$</div></li></ol>

Deriving Inverse Trigonometric Derivatives

Techniques of Differentiation

<p>Show that</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}{{\sin }^{-1}}x=\frac{1}{\sqrt{1-{{x}^{2}}}}$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}{{\tan }^{-1}}x=\frac{1}{1+{{x}^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}{{\cos }^{-1}}x=-\frac{1}{\sqrt{1-{{x}^{2}}}}$</div></li></ol>

<ol class="i-list"><li data-list-item-id="item-1759480976134-qvs8tdu94" data-list-item-path="item-1759480976134-qvs8tdu94"><div class="list-item-paragraph">Prove that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}\left( {{\tan }^{-1}}x \right)=\frac{1}{1+{{x}^{2}}}$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The diagram shows a sketch of the graph of $y=\frac{1}{1+{{x}^{2}}}$.</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/1772534495974-fz9hkppcq1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzI1MzQ0OTU5NzQtZno5aGtwcGNxMS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzg5MzA2LCJleHAiOjE3OTI3ODk2MDZ9.BbTsgLxiQZyJAQKU4GzZ8tsRAWlvGn_YBqgby7om1I4" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list" start="2"><li data-list-item-id="item-1759481010866-tnjs4vxfr" data-list-item-path="item-1759481010866-tnjs4vxfr"><div class="list-item-paragraph">By considering the shaded rectangle and the area of the region between the graph and the $x$-axis for&nbsp;$n-1\le x\le n$, where&nbsp;$n\ge 1$, show that</div><div class="list-item-paragraph" style="text-align: center">$\frac{1}{1+{{n}^{2}}}&lt;{{\tan }^{-1}}\left( n \right)-{{\tan }^{-1}}\left( n-1 \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759481014053-poqay0l4i" data-list-item-path="item-1759481014053-poqay0l4i"><div class="list-item-paragraph">Deduce that&nbsp;$\frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\ldots +\frac{1}{1+{{n}^{2}}}&lt;{{\tan }^{-1}}\left( n \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The region $R$ is bounded by the curve $y=\frac{1}{1+{{x}^{2}}}$ and the lines $x=3$ and $y=\frac{1}{2}$. Write down the equation of the curve obtained when the curve $y=\frac{1}{1+{{x}^{2}}}$ is translated by $-3$ units in the direction of the $x$-axis. Hence, or otherwise, find the volume of the solid formed when&nbsp;$R$ is rotated completely about the line $x=3$.<span class="inline-marks">[4]</span></p>

Prove that $\frac{\mathrm{d}}{\mathrm{d}x}\left( {{\tan }^{-1}}x \right)=\frac{1}{1+{{x}^{2}}}$.

The diagram shows a sketch of the graph of $y=\frac{1}{1+{{x}^{2}}}$.

By considering the shaded rectangle and the area of the region between the graph and the $x$-axis for $n-1\le x\le n$, where $n\ge 1$, show that

Deduce that $\frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\ldots +\frac{1}{1+{{n}^{2}}}<{{\tan }^{-1}}\left( n \right)$.

The region $R$ is bounded by the curve $y=\frac{1}{1+{{x}^{2}}}$ and the lines $x=3$ and $y=\frac{1}{2}$. Write down the equation of the curve obtained when the curve $y=\frac{1}{1+{{x}^{2}}}$ is translated by $-3$ units in the direction of the $x$-axis. Hence, or otherwise, find the volume of the solid formed when $R$ is rotated completely about the line $x=3$.

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