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