Derivative of cos(cos(x))

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cos(cos(x))

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

cos(cos(x))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                        2               
cos(x)*sin(cos(x)) - sin (x)*cos(cos(x))

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of cos(cos(x))

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