# Differentiate sin(sin(x)) with respect to x ?

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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))

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

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

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