Correct Answer - Option 4 :
\(\rm\sin(cos(x))(sin(x))\)
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 = cos(cos(x))
⇒ \(\rm \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} }{\mathrm{d} x}cos(cos(x))\)
⇒ \(\rm \frac{\mathrm{d} y}{\mathrm{d} x}= \ -\sin(cos(x))\frac{\mathrm{d} }{\mathrm{d} x}(cos(x))\)
⇒ \(\rm \frac{\mathrm{d} y}{\mathrm{d} x}=-sin(cos(x))(-sin(x))\)
⇒ \(\rm \frac{\mathrm{d} y}{\mathrm{d} x}=sin(cos(x))(sin(x))\)