Question

44 points ✌✌

Differentiate
${ \sin}^{2} ( {x)}^{2} w.r.t. {x}^{2} .$
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Answer 1

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Answer 2

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Let

$y = { \sin }^{2} {x }^{2} \: \: and \: z = {x}^{2}$

$therefore \: \frac{dy}{dx} = 2 \sin {x}^{2} \cos {x}^{2} \times \\ 2x = 2x \sin( {2x}^{2} ) \: and \: \frac{dz}{dx} = 2x$

Now

$\frac{dy}{dz} = \frac{dy}{dx} \times \frac{dx}{dz} = 2x \sin ({2x)}^{2} \times \\ \frac{1}{2x} = \sin( {2x}^{2} ) \: or \: 2 \sin( {x}^{2} ) \cos( {x}^{2} )$

Alternative

$\frac{d}{ {dx}^{2} } ( { { \sin }^{2} ( {x}^{2} ) })^{2} = 2 \sin( {x}^{2} ) \times \frac{d}{ {dx}^{2} } \\ ( \sin( {x}^{2} ) ) = 2 \sin( {x}^{2} ) \times \cos( {x}^{2} ) \frac{d}{ {dx}^{2} }( {x}^{2} )$

$= 2 \sin( {x}^{2} ) \cos( {x}^{2} ) \times 1 = \\ \sin( {2x}^{2} ) .$