Differentiate cos(cos(x)) with respect to x ?

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Agniveer Vayu Science (Group X) 22nd March 2025 Memory-Based Paper
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  1. \(\rm\sin(sin(x))(sin(x))\)
  2. \(\rm\sin(sin(x))(cos(x))\)
  3. \(\rm\cos(sin(x))(sin(x))\)
  4. \(\rm\sin(cos(x))(sin(x))\)

Answer (Detailed Solution Below)

Option 4 : \(\rm\sin(cos(x))(sin(x))\)
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Detailed Solution

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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))

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

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

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

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

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