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If $y= e^{\sin^{2}x}$ ; dy/dx equals

Option 1)
$e^{\cos ^{2 }x}$

Option 2)
$e^{\sin 2x}$

Option 3)
$e^{\sin^{2}x \cdot \sin 2x}$

Option 4)
$e^{\sin 2x }\cdot 2 \cos 2x$

Answers (1)

best_answer

As we have learned

Chain Rule for differentiation (indirect) -

Let y = f(x) is not in standard form then

$\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$

$ex:\:\:y=sin(ax+b)$

$Let\:\;u=(ax+b)$

$then\:\:y=sin \:u$

$so\:\:\frac{dy}{du}=cos \:u\:\:and\:\:\frac{du}{dx}=a$

$\therefore \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}=a\:cos \:u$

$=a\:cos(ax+b)$

- wherein

$Where\;\:y=f(u)\:\;and\;\;u=f(x)$

$\frac{dy}{dx} =\frac{dy}{dx}\cdot \frac{du}{dx}$

Let $u= \sin ^{2}x , v= \sin x$ then u= $v^{2}$ and $y=e^{u}$

$\Rightarrow \frac{dy}{dx} = \frac{de^{u}}{du}*\frac{dv^{2}}{dv}*\frac{d(\sin x)}{dx}$

$\Rightarrow \frac{dy}{dx} = e^{u}2v* \cos x =e^{\sin ^{2}x} * 2\sin x*\cos x$

$\Rightarrow \frac{dy}{dx} = e^{\sin ^{2}x}\sin 2x$

Option 1)

$e^{\cos ^{2 }x}$

Option 2)

$e^{\sin 2x}$

Option 3)

$e^{\sin^{2}x \cdot \sin 2x}$

Option 4)

$e^{\sin 2x }\cdot 2 \cos 2x$

Posted by

Himanshu

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