Question

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$x = \frac{ {sin}^{3}t }{ \sqrt{cos2t}}$
$y = \frac{cos {}^{3} t}{ \sqrt{cos2t} }$
Find dy /dx ​

Answer 1

Answer:

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Answer 2

Answer:

$\large -\cot 3 t .$

Step-by-step explanation:

Given :

$\large \text{ x=\dfrac{\sin^3t}{\sqrt{\cos2t}} }$

$\large \text{ y=\dfrac{\cos^3t}{\sqrt{\cos2t}} }$

We have to find $\large \dfrac{dy}{dx}$

Here apply both side d t

$\large \implies\dfrac{dy}{dx}\times\dfrac{dt}{dt}\\\\\\\large \implies\dfrac{dy}{dt}\times\dfrac{dt}{dx}\\\\\\\large \text{ \implies\dfrac{\dfrac{dy}{dt}}{\dfrac{dt}{dx}} }$

First let us find d y / d t

$\displaystyle \text{ \dfrac{dy}{dx} =\dfrac{d}{dt}\left ({\dfrac{\sin^3t}{\sqrt{\cos2t}}}\right) }$

Applying division rule of differentation here

$\displaystyle \text{ \dfrac{d}{dx}\left(\frac{fx}{gx}\right)=\dfrac{gx\dfrac{d}{dt}\left( fx \right)-fx\dfrac{d}{dt}\left(gx\right)}{gx^2} }$

$\displaystyle \text{ \dfrac{d}{dx}\left(\frac{cos^3t}{\sqrt{cos2t}}\right)=\dfrac{\sqrt{cos2t} \ \dfrac{d}{dt}\left( cos^3t \right)-cos^3t \ \dfrac{d}{dt}\left(\sqrt{cos2t}\right)}{(\sqrt{cos2t})^2} }\\\\\\\large \text{On solving this we get}\\\\\\ \displaystyle \text{ \dfrac{d}{dx}\left(\frac{cos^3t}{\sqrt{cos2t}}\right)=\dfrac{\cos^2t(-3\sin t .\cos2 t + \cos t . \sin 2t)}{(cos2t)^{3/2}} }$

Now for d x / d t

$\displaystyle \text{ \dfrac{d}{dx}\left(\frac{\sin^3t}{\sqrt{\cos2t}}\right)=\dfrac{\sqrt{\cos2t} \ \dfrac{d}{dt}\left( \sin^3t \right)-\sin^3t \ \dfrac{d}{dt}\left(\sqrt{\cos2t}\right)}{(\sqrt{\cos2t})^2} }\\\\\\\large \text{On solving this we get}\\\\\\ \displaystyle \text{ \dfrac{d}{dx}\left(\frac{\sin^3t}{\sqrt{cos2t}}\right)=\dfrac{\sin^2t(3\cos t .\cos2 t + \sin t . \sin 2t)}{(cos2t)^{3/2}} }$

$\displaystyle \text{Now for \dfrac{\dfrac{dy}{dt} }{\dfrac{dx}{dt}} }$

$\displaystyle \text{ \dfrac{\dfrac{dy}{dt} }{\dfrac{dx}{dt} }=\frac{\dfrac{\cos^2t(-3\sin t .\cos2 t + \cos t . \sin 2t)}{(cos2t)^{3/2}} }{\dfrac{\sin^2t(3\cos t .\cos2 t + \sin t . \sin 2t)}{(cos2t)^{3/2}}} }\\\\\\ \displaystyle \dfrac{dy}{dx}=\frac{ \cos^2t(-3\sin t .\cos2 t + \cos t . \sin 2t}{\sin^2t(3\cos t .\cos2 t + \sin t . \sin 2t)}\\\\\\\large \text{On solving this we get}\\\\\\\displaystyle \dfrac{dy}{dx}=-\cot 3t$

Thus we get answer.