Derivative of y=ln^5arctg1/x

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   5         
log (atan(1))
-------------
      x

$$\frac{\log{\left(\operatorname{atan}{\left(1 \right)} \right)}^{5}}{x}$$

  /   5         \
d |log (atan(1))|
--|-------------|
dx\      x      /

$$\frac{d}{d x} \frac{\log{\left(\operatorname{atan}{\left(1 \right)} \right)}^{5}}{x}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
So, the result is: $- \frac{\log{\left(\operatorname{atan}{\left(1 \right)} \right)}^{5}}{x^{2}}$
Now simplify:

$- \frac{\log{\left(\frac{\pi}{4} \right)}^{5}}{x^{2}}$

The answer is:

$- \frac{\log{\left(\frac{\pi}{4} \right)}^{5}}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    5          
-log (atan(1)) 
---------------
        2      
       x

$$- \frac{\log{\left(\operatorname{atan}{\left(1 \right)} \right)}^{5}}{x^{2}}$$

Simplify

The second derivative [src]

     5         
2*log (atan(1))
---------------
        3      
       x

$$\frac{2 \log{\left(\operatorname{atan}{\left(1 \right)} \right)}^{5}}{x^{3}}$$

Simplify

The third derivative [src]

      5         
-6*log (atan(1))
----------------
        4       
       x

$$- \frac{6 \log{\left(\operatorname{atan}{\left(1 \right)} \right)}^{5}}{x^{4}}$$

Simplify

The graph