If $f(x)=x+1$, then write the value of
Question: If $f(x)=x+1$, then write the value of Solution: $f(x)=x+1$ $\Rightarrow(fof)(x)=f(x)+1$ $=(x+1)+1$ $=x+2$ So, $\frac{\mathrm{d}}{\mathrm{dx}}($ fof $)(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x}+2)$ $=1($ Ans $)$...
Read More →If $f(x)=log _{e}left(log _{e} x ight)$, then write the value of $f^{prime}(e)$.
Question: If $f(x)=\log _{e}\left(\log _{e} x\right)$, then write the value of $f^{\prime}(e)$. Solution: $f(x)=\log _{e}\left(\log _{e} x\right)$ Using the Chain Rule of Differentiation, $f(x)=\frac{1}{\log _{e} x} \cdot \frac{1}{x}$ So, $f(e)=\frac{1}{\log _{e} e} \cdot \frac{1}{e}$ $=\frac{1}{e}$ (Ans)...
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Question: Choose the correct alternative in the following: If $y=\tan ^{-1}\left(\frac{\sin x+\cos x}{\cos x-\sin x}\right)$, then $\frac{d y}{d x}$ is equal to A. $\frac{1}{2}$ B. 0 C. 1 D. none of these Solution: $y=\tan ^{-1}\left(\frac{\sin x+\cos x}{\cos x-\sin x}\right)$ Dividing Numerator and denominator by $\cos x$ we get, $y=\tan ^{-1}\left(\frac{\frac{\sin x}{\cos x}+\frac{\cos x}{\cos x}}{\frac{\cos x}{\cos x}-\frac{\sin x}{\cos x}}\right)$ $y=\tan ^{-1}\left(\frac{\tan x+1}{1-1 \cdot...
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Question: Choose the correct alternative in the following: If $y=\sqrt{\sin x+y}$, then $\frac{d y}{d x}$ equals. A. $\frac{\cos x}{2 y-1}$ B. $\frac{\cos x}{1-2 y}$ c. $\frac{\sin x}{1-2 y}$ D. $\frac{\sin x}{2 y-1}$ Solution: $y=\sqrt{\sin x+y}$ Squaring both sides, we get $y^{2}=\sin x+y$ Differentiating w.r.t y we get $2 y=\cos x \frac{d x}{d y}+1$ $\frac{\mathrm{dx}}{\mathrm{dy}}=\frac{2 \mathrm{y}-1}{\cos \mathrm{x}}$ $\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\cos \mathrm{x}}{2 \ma...
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Question: Choose the correct alternative in the following: If $y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)$, then $\frac{d y}{d x}=$ A. $\frac{4 \mathrm{x}^{3}}{1-\mathrm{x}^{4}}$ B. $-\frac{4 x}{1-x^{4}}$ C. $\frac{1}{4-x^{4}}$ D. $\frac{4 x^{3}}{1-x^{4}}$ Solution: $y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)$ $\frac{d y}{d x}=\frac{1}{\frac{1-x^{2}}{1+x^{2}}}\left[\frac{\frac{d\left(1-x^{2}\right)}{d x} \cdot\left(1+x^{2}\right) \frac{d\left(1+x^{2}\right)}{d x} \cdot\left(1-x^{2}\right)}{\le...
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Question: Choose the correct alternative in the following: If $\sin y=x \cos (a+y)$, then $\frac{d y}{d x}$ is equal to A. $\frac{\cos ^{2}(a+y)}{\cos a}$ B. $\frac{\cos a}{\cos ^{2}(a+y)}$ C. $\frac{\sin ^{2} y}{\cos a}$ D. none of these Solution: $\sin y=x \cos (a+y)$ $x=\frac{\sin y}{\cos (a+y)}$ Differentiating w.r.t y we get, $\frac{d x}{d y}=\frac{\frac{d \sin y}{d x} \cdot \cos (a+y)-\frac{d \cos (a+y)}{d x} \cdot(\sin y)}{\cos ^{2}(a+y)}($ Using quotient rule $)$ $\frac{d x}{d y}=\frac{\...
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Question: Choose the correct alternative in the following: If $\sin ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\log$ a then $\frac{d y}{d x}$ is equal to A. $\frac{\mathrm{x}^{2}-\mathrm{y}^{2}}{\mathrm{x}^{2}+\mathrm{y}^{2}}$ B. $\frac{\mathrm{y}}{\mathrm{x}}$ C. $\frac{\mathrm{x}}{\mathrm{y}}$ D. none of these Solution: $\sin ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\log a$ $\frac{x^{2}-y^{2}}{x^{2}+y^{2}}=\sin (\log a)$ Put $y=x \tan \theta$ $\theta=\tan ^{-1}\left(\frac{y}{x}...
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Question: Choose the correct alternative in the following: If $\mathrm{y}=\log \sqrt{\tan \mathrm{x}}$, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\frac{\pi}{4}$ is given by A. $\infty$ B. 1 C. 0 D. $1 / 2$ Solution: $\mathrm{y}=\log \sqrt{\tan \mathrm{x}}$ $\Rightarrow \mathrm{y}=\log (\tan \mathrm{x})^{\frac{1}{2}}$ $\Rightarrow \mathrm{y}=\frac{1}{2} \log (\tan \mathrm{x})$ Differentiating w.r.t $x$ we get, $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2} \cdot...
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Question: Choose the correct alternative in the following: If $\sqrt{1-x^{6}}+\sqrt{1-y^{6}}=a^{3}\left(x^{3}-y^{3}\right)$, then $\frac{d y}{d x}$ is equal to A. $\frac{\mathrm{x}^{2}}{\mathrm{y}^{2}} \sqrt{\frac{1-\mathrm{y}^{6}}{1-\mathrm{x}^{6}}}$ B. $\frac{y^{2}}{x^{2}} \sqrt{\frac{1-y^{6}}{1-x^{6}}}$ C. $\frac{x^{2}}{y^{2}} \sqrt{\frac{1-x^{6}}{1-y^{6}}}$ D. none of these Solution: $\sqrt{1-x^{6}}+\sqrt{1-y^{6}}=a^{3}\left(x^{3}-y^{3}\right)$ Let $x^{3}=\cos p$ and $y^{3}=\cos q$ $\cos ^{-...
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Question: Choose the correct alternative in the following: If, $y=\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}} \cdot+\frac{1}{1+x^{b-a}+x^{c-a}}$, then $\frac{d y}{d x}$ is equal to A. 1 B. $(a+b-c)^{x^{a+b+c-1}}$ C. 0 D. none of these Solution: $y=\frac{1}{1+x^{2-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}+\frac{1}{1+x^{b-a}+x^{c-a}}$ $\Rightarrow y=\frac{1}{1+\frac{x^{2}}{x^{b}}+\frac{x^{c}}{x^{b}}}+\frac{1}{1+\frac{x^{b}}{x^{c}}+\frac{x^{a}}{x^{c}}}+\frac{1}{1+\frac{x^{b}}{x^{a}}+\frac...
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Question: Choose the correct alternative in the following: If $f(x)=\left(\frac{x^{1}}{x^{m}}\right)^{1+m}\left(\frac{x^{m}}{x^{n}}\right)^{m+n}\left(\frac{x^{n}}{x^{1}}\right)^{n+1}$, then $f^{\prime}(x)$ is equal to A. 1 B. 0 C. $x^{l+m+n}$ D. none of these Solution: $f(x)=\left(\frac{x^{1}}{x^{m}}\right)^{1+m}\left(\frac{x^{m}}{x^{n}}\right)^{m+n}\left(\frac{x^{n}}{x^{1}}\right)^{n+1}$ $f(x)=\frac{\left(x^{1}\right)^{1+m} \cdot\left(x^{m}\right)^{m+n} \cdot\left(x^{n}\right)^{n+1}}{\left(x^{m...
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Question: Choose the correct alternative in the following: If $f(x)=\left(\frac{x^{1}}{x^{m}}\right)^{1+m}\left(\frac{x^{m}}{x^{n}}\right)^{m+n}\left(\frac{x^{n}}{x^{1}}\right)^{n+1}$, then $f^{\prime}(x)$ is equal to A. 1 B. 0 C. $x^{l+m+n}$ D. none of these Solution: $f(x)=\left(\frac{x^{1}}{x^{m}}\right)^{1+m}\left(\frac{x^{m}}{x^{n}}\right)^{m+n}\left(\frac{x^{n}}{x^{1}}\right)^{n+1}$ $f(x)=\frac{\left(x^{1}\right)^{1+m} \cdot\left(x^{m}\right)^{m+n} \cdot\left(x^{n}\right)^{n+1}}{\left(x^{m...
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Question: Choose the correct alternative in the following: If $f(x)=|x-3|$ and $g(x)=$ fof $(x)$, then for $x10, g^{\prime}(x)$ is equal to A. 1 B. $-1$ C. 0 D. none of these Solution: $g(x)=f \circ f(x)=f(f(x))=|f(x)-3| \because f(x)=|x-3|$ $=|| x-3|-3|$ $\because|x-3|=\left\{\begin{array}{c}(x-3), x3 \\ -(x-3), x3\end{array}\right.$ Since we have given $x10$ then $|x-3|=(x-3)$ $\therefore g(x)=|(x-3)-3|=|x-6|$ $\because|x-6|=\left\{\begin{array}{l}(x-6), x6 \\ -(x-6), x6\end{array}\right.$ Sin...
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Question: Choose the correct alternative in the following: If $f(x)=\sqrt{x^{2}-10 x+25}$, then the derivative of $f(x)$ in the interval $[0,7]$ is A. 1 B. $-1$ C. 0 D. none of these Solution: $f(x)=\sqrt{x^{2}-10 x+25}$ $\Rightarrow f(x)=\sqrt{x^{2}-(2)(5) x+5^{2}}$ $\Rightarrow f(x)=\sqrt{(x-5)^{2}}$ $\Rightarrow f(x)=|x-5|$ $\Rightarrow f(x)=\left\{\begin{array}{c}(x-5), x-5 \geq 0 \Leftrightarrow x \geq 5 \\ -(x-5), x-50 \Leftrightarrow x5\end{array}\right.$ $\Rightarrow \mathrm{f}^{\prime}(...
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Question: Choose the correct alternative in the following: If $f(x)=\left|x^{2}-9 x+20\right|$, then $f^{\prime}(x)$ is equal to A. $-2 x+9$ for all $x \in R$ B. $2 x-9$ if $4x5$ C. $-2 x+9$ if $4x5$ D. none of these Solution: $f(x)=\left|x^{2}-9 x+20\right|$ $=\left|x^{2}-4 x-5 x+20\right|$ $=|x(x-4)-5(x-4)|$ $f(x)=|(x-5)(x-4)|$ $\Rightarrow \mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}(\mathrm{x}-5)(\mathrm{x}-4), \mathrm{x} \geq 5 \text { and } \mathrm{x} \geq 4 \\ -(\mathrm{x}-5)(\mathrm{x}...
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Question: Choose the correct alternative in the following: If $f(x)=\sqrt{x^{2}+6 x+9}$, then $f^{\prime}(x)$ is equal to A. 1 for $x-3$ B. $-1$ for $x-3$ C. 1 for all $x \in R$ D. none of these Solution: $f(x)=\sqrt{x^{2}+6 x+9}$ $\Rightarrow f(x)=\sqrt{(x+3)^{2}}$ $\Rightarrow f(x)=|x+3|$ $\Rightarrow \mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}(\mathrm{x}+3), \mathrm{x}+3 \geq 0 \Leftrightarrow \mathrm{x} \geq-3 \\ -(\mathrm{x}+3), \mathrm{x}+30 \Leftrightarrow \mathrm{x}-3\end{array}\right...
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Question: Choose the correct alternative in the following: The derivative of $\cos ^{-1}\left(2 x^{2}-1\right)$ with respect to $\cos ^{-1} x$ is A. 2 B. $\frac{1}{2 \sqrt{1-x^{2}}}$ C. $2 / \mathrm{x}$ D. $1-x^{2}$ Solution: Let $u=\cos ^{-1}\left(2 x^{2}-1\right)$ and $v=\cos ^{-1} x$ $\frac{\mathrm{du}}{\mathrm{dv}}=?$ Considering $u=\cos ^{-1}\left(2 x^{2}-1\right)$ Put $x=\cos \theta \Rightarrow \theta=\cos ^{-1} x \cdots$ (1) $u=\cos ^{-1}\left(2 \cos ^{2} \theta-1\right)$ $u=\cos ^{-1}(\c...
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Question: Choose the correct alternative in the following: If $\sin y=x \sin (a+y)$, then $\frac{d y}{d x}$ is A. $\frac{\sin a}{\sin a \sin ^{2}(a+y)}$ B. $\frac{\sin ^{2}(a+y)}{\sin a}$ C. $\sin a \sin ^{2}(a+y)$ D. $\frac{\sin ^{2}(a-y)}{\sin a}$ Solution: $\sin y=x \sin (a+y)$ $\Rightarrow \frac{\sin y}{\sin (a+y)}=x$ Differentiating w.r.t y we get, $\Rightarrow \frac{d x}{d y}=\frac{d}{d y}\left(\frac{\sin y}{\sin (a+y)}\right)$ $\Rightarrow \frac{d x}{d y}=\frac{d}{d y}\left(\frac{\sin y}{...
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Question: Choose the correct alternative in the following: If $3 \sin (x y)+4 \cos (x y)=5$, then $\frac{d y}{d x}=$ A. $-\frac{\mathrm{y}}{\mathrm{x}}$ B. $\frac{3 \sin (x y)+4 \cos (x y)}{3 \cos (x y)-4 \sin (x y)}$ C. $\frac{3 \cos (\mathrm{xy})+4 \sin (\mathrm{xy})}{4 \cos (\mathrm{xy})-3 \sin (\mathrm{xy})}$ D. none of these Solution: $3 \sin (x y)+4 \cos (x y)=5$ Differentiating w.r.t $x$ we get, $\Rightarrow 3\left[\cos (\mathrm{xy}) \cdot\left(1 \cdot \mathrm{y}+\mathrm{x} \cdot \frac{\m...
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Question: Choose the correct alternative in the following: If $y=\sqrt{\sin x+y}$, then $\frac{d y}{d x}=$ A. $\frac{\sin x}{2 y-1}$ B. $\frac{\sin x}{1-2 y}$ C. $\frac{\cos x}{1-2 y}$ D. $\frac{\cos x}{2 y-1}$ Solution: $y=\sqrt{\sin x+y}$ Squaring both sides $\Rightarrow y^{2}=\sin x+y$ Differentiating w.r.t $x$ we get, $\Rightarrow 2 y \cdot \frac{d y}{d x}=\cos x+\frac{d y}{d x}$ $\Rightarrow \frac{d y}{d x}(2 y-1)=\cos x$ $\Rightarrow \frac{d y}{d x}=\frac{\cos x}{2 y-1}=D$...
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Question: Choose the correct alternative in the following: $\frac{d}{d x}\left[\log \left\{e^{x}\left(\frac{x-2}{x+2}\right)^{3 / 4}\right\}\right]$ equals A. $\frac{\mathrm{x}^{2}-1}{\mathrm{x}^{2}-4}$ B. 1 C. $\frac{x^{2}+1}{x^{2}-4}$ D. $e^{x} \frac{x^{2}-1}{x^{2}-4}$ Solution: $\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left\{\mathrm{e}^{\mathrm{x}}\left(\frac{\mathrm{x}-2}{\mathrm{x}+2}\right)^{\frac{2}{4}}\right\}\right]$ Let $u=\frac{x-2}{x+2} \Rightarrow \frac{d u}{d x}=\frac{1 \cdot(x+2)...
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Question: Choose the correct alternative in the following: $\frac{d}{d x}\left[\log \left\{e^{x}\left(\frac{x-2}{x+2}\right)^{3 / 4}\right\}\right]$ equals A. $\frac{\mathrm{x}^{2}-1}{\mathrm{x}^{2}-4}$ B. 1 C. $\frac{x^{2}+1}{x^{2}-4}$ D. $e^{x} \frac{x^{2}-1}{x^{2}-4}$ Solution: $\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left\{\mathrm{e}^{\mathrm{x}}\left(\frac{\mathrm{x}-2}{\mathrm{x}+2}\right)^{\frac{2}{4}}\right\}\right]$ Let $u=\frac{x-2}{x+2} \Rightarrow \frac{d u}{d x}=\frac{1 \cdot(x+2)...
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Question: Choose the correct alternative in the following: $\frac{\mathrm{d}}{\mathrm{dx}}\left\{\tan ^{-1}\left(\frac{\cos \mathrm{x}}{1+\sin \mathrm{x}}\right)\right\}$ equals A. $1 / 2$ B. $-1 / 2$ C. 1 D. $-1$ Solution: $\frac{d}{d x}\left\{\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)\right\}$ $\Rightarrow \frac{d}{d x}\left\{\tan ^{-1}\left(\frac{\sin \left(\frac{\pi}{2}-x\right)}{1+\cos \left(\frac{\pi}{2}-x\right)}\right)\right\} \because \sin \left(\frac{\pi}{2}-x\right)=\cos x$ and $\...
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Question: Choose the correct alternative in the following: Let $U=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ and $V=\tan ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$, then $\frac{d U}{d V}=$ A. $1 / 2$ B. $x$ C. $\frac{1-\mathrm{x}^{2}}{1+\mathrm{x}^{2}}$ D. 1 Solution: We are given that $U=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right), V=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$ $\frac{\mathrm{dU}}{\mathrm{dV}}=?$ Now, we know $\frac{\mathrm{dU}}{\mathrm{dV}}=\frac{\frac{\mathrm{dU}}{\mathrm{dx}}}{\f...
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Question: Choose the correct alternative in the following: If $\sin (x+y)=\log (x+y)$, then $\frac{d y}{d x}=$ A. 2 B. $-2$ C. 1 D. $-1$ Solution: $\sin (x+y)=\log (x+y)$ Differentiating w.r.t $\mathrm{x}$ we get, $\Rightarrow \cos (x+y) \cdot\left(1+\frac{d y}{d x}\right)=\frac{1}{x+y} \cdot\left(1+\frac{d y}{d x}\right)$ $\Rightarrow \cos (x+y) \cdot\left(1+\frac{d y}{d x}\right)-\frac{1}{x+y} \cdot\left(1+\frac{d y}{d x}\right)=0$ $\Rightarrow\left(1+\frac{\mathrm{dy}}{\mathrm{dx}}\right)\lef...
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