Derivative of cos(log^2x)

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   /   2   \
cos\log (x)/

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

cos(log(x)^2)

Detail solution

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

$- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} \right)}}{x}$

The answer is:

$- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             /   2   \
-2*log(x)*sin\log (x)/
----------------------
          x

$$- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} \right)}}{x}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /     /   2   \        /   2   \                      /   2   \        3       /   2   \        2       /   2   \\
2*\3*sin\log (x)/ - 6*cos\log (x)/*log(x) - 2*log(x)*sin\log (x)/ + 4*log (x)*sin\log (x)/ + 6*log (x)*cos\log (x)//
--------------------------------------------------------------------------------------------------------------------
                                                          3                                                         
                                                         x

$$\frac{2 \left(4 \log{\left(x \right)}^{3} \sin{\left(\log{\left(x \right)}^{2} \right)} + 6 \log{\left(x \right)}^{2} \cos{\left(\log{\left(x \right)}^{2} \right)} - 2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} \right)} - 6 \log{\left(x \right)} \cos{\left(\log{\left(x \right)}^{2} \right)} + 3 \sin{\left(\log{\left(x \right)}^{2} \right)}\right)}{x^{3}}$$

Simplify

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Identical expressions

Derivative of cos(log^2x)

The graph:

Piecewise:

The solution