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Derivative of:
Derivative of e^(x^2)
Derivative of 11
Derivative of -5
Derivative of sen(x)
Identical expressions
cos^ two *(lnx)
co sinus of e of squared multiply by (lnx)
co sinus of e of to the power of two multiply by (lnx)
cos2*(lnx)
cos2*lnx
cos²*(lnx)
cos to the power of 2*(lnx)
cos^2(lnx)
cos2(lnx)
cos2lnx
cos^2lnx

The derivative of the function/ cos^2*(lnx)

Derivative of cos^2*(lnx)

The solution

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   2        
cos (log(x))

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

d /   2        \
--\cos (log(x))/
dx

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of cos^2*(lnx)

The graph:

Piecewise:

The solution