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cos(cosx)^2

The derivative of the function/ cos(cos(x))^2

Derivative of cos(cos(x))^2

The solution

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   2        
cos (cos(x))

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

cos(cos(x))^2

Detail solution

Let $u = \cos{\left(\cos{\left(x \right)} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(\cos{\left(x \right)} \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. 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 result of the chain rule is:

$2 \sin{\left(x \right)} \sin{\left(\cos{\left(x \right)} \right)} \cos{\left(\cos{\left(x \right)} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                2                       2                       2                           \       
2*\-cos(cos(x))*sin(cos(x)) - 3*cos (cos(x))*cos(x) + 3*sin (cos(x))*cos(x) - 4*sin (x)*cos(cos(x))*sin(cos(x))/*sin(x)

$$2 \left(- 4 \sin^{2}{\left(x \right)} \sin{\left(\cos{\left(x \right)} \right)} \cos{\left(\cos{\left(x \right)} \right)} + 3 \sin^{2}{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} - \sin{\left(\cos{\left(x \right)} \right)} \cos{\left(\cos{\left(x \right)} \right)} - 3 \cos{\left(x \right)} \cos^{2}{\left(\cos{\left(x \right)} \right)}\right) \sin{\left(x \right)}$$

Simplify

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Derivative of cos(cos(x))^2

The graph:

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The solution