Question

diffrentiate with raspect to x. y=sin(m sin x)

Answer 1

Hope u like my process
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Suppose

=> a function f of z represents y

Whereas, if z is represented by g of (x)

For example..

$= >y = f(z) \\ \\ then \: \: if \: \: z = g(x) \\ \\ then \: \bf \: \: \: \: y = f(z) = f( g(x))$

Now the differentiation too takes place in a way such that..

$= > \boxed{ \bf \frac{dy}{dx} = \green{\frac{df}{dz} \times \frac{dz}{dx} }}$
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Coming to the solving part.. Of the given question


=> y = sin( m sin (x))

Let m sinx be z

So..

=> y = sin z

$= > \frac{dy}{dx} = \blue{ \frac{d( \sin(z) )}{dz} \times \frac{d(z)}{dx}} \\ \\ = > \frac{dy}{dx} = \blue{ \cos(z) \times \frac{d(m \: \sin(x) )}{dx} } \\ \\ = > \frac{dy}{dx} = \blue{ \cos(m \sin(x) ) \times m \cos(x) } \\ \\ = > \boxed{ \bf \: \frac{dy}{dx} = \underline{\orange{m \cos(x) .\cos(m \sin(x) ) } }}$
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Hope this is ur required answer

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