Derivative of sin(sin(x))

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sin(sin(x))

$$\sin{\left(\sin{\left(x \right)} \right)}$$

sin(sin(x))

Detail solution

Let $u = \sin{\left(x \right)}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result of the chain rule is:

$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$

The answer is:

$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)*cos(sin(x))

$$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$$

Simplify

The second derivative [src]

 /   2                                    \
-\cos (x)*sin(sin(x)) + cos(sin(x))*sin(x)/

$$- (\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)})$$

Simplify

The third derivative [src]

/                  2                                      \       
\-cos(sin(x)) - cos (x)*cos(sin(x)) + 3*sin(x)*sin(sin(x))/*cos(x)

$$\left(3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}$$

Simplify

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Derivative of sin(sin(x))

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