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Differentiate sin(sin(x)) with respect to x ?
1. \(\rm\cos(sin(x))(cos(x))\)
2. \(\rm\sin(sin(x))(cos(x))\)
3. \(\rm\cos(sin(x))(sin(x))\)
4. \(\rm\cos(cos(x))(cos(x))\)

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Correct Answer - Option 1 : \(\rm\cos(sin(x))(cos(x))\)

Concept:

Chain rule:

Let y = f(v) be a differentiable function of v and v = g(x) be a differentiable function of x then \(\frac{{dy}}{{dx}} = \frac{{dy}}{{dv}} ⋅ \frac{{dv}}{{dx}}\)

Calculation:

Let y = sin(sin(x))

⇒ \(\rm \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} }{\mathrm{d} x}sin(sin(x))\)

⇒ \(\rm \frac{\mathrm{d} y}{\mathrm{d} x}=cos(sin(x))\frac{\mathrm{d} }{\mathrm{d} x}(sin(x))\)

⇒ \(\rm \frac{\mathrm{d} y}{\mathrm{d} x}=cos(sin(x))(cos(x))\)

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