Question

Assertion :$u=f\left(\cot x\right)$ & $v=g\left(\text{cosec}x\right)$ & $f^{\prime }\left(1\right)=\sqrt{2}$ and $g^{\prime }\left(\sqrt{2}\right)=2$ then ${\left(\frac{du}{dv}\right)}_{x=\frac{\pi }{4}}=1$ Reason: If $u=f\left(x\right),v=g\left(x\right)$ then derivative of $f$ w.r.t. to $g$ is $\frac{du}{dv}=\frac{du/dx}{dv/dx}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$u=f\left(\cot x\right)$, $v=g\left(cosecx\right)$
$\Rightarrow$ $\frac{du}{dx}=f^{\prime }\left(\cot x\right)(-cose{c}^{2}x)$ & $\frac{dv}{dx}=g^{\prime }\left(cosecx\right)(-cosecx.\cot x)$
$\therefore$
$\frac{du}{dv}=\frac{\frac{du}{dx}}{\frac{dv}{dx}}=\frac{f^{\prime }\left(\cot x\right)(-cose{c}^{2}x)}{g^{\prime }\left(cosecx\right)(-cosecx.\cot x)}$

$=\frac{f^{\prime }\left(\cot x\right)}{g^{\prime }\left(cosecx\right)}.\frac{1}{\cos x}$

$\therefore$ ${\left(\frac{du}{dv}\right)}_{x=\frac{\pi }{4}}=\frac{f^{\prime }\left(1\right)}{g^{\prime }\left(\sqrt{2}\right)}\sqrt{2}=\frac{\left(\sqrt{2}\right)\left(\sqrt{2}\right)}{2}=1$