Derivative of ln^4(x/5)

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   4/x\
log |-|
    \5/

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

d /   4/x\\
--|log |-||
dx\    \5//

$$\frac{d}{d x} \log{\left(\frac{x}{5} \right)}^{4}$$

Transform

Detail solution

Let $u = \log{\left(\frac{x}{5} \right)}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\frac{x}{5} \right)}$ :
1. Let $u = \frac{x}{5}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{5}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $\frac{1}{5}$
  The result of the chain rule is:
  
  $\frac{1}{x}$
The result of the chain rule is:

$\frac{4 \log{\left(\frac{x}{5} \right)}^{3}}{x}$
Now simplify:

$\frac{4 \log{\left(\frac{x}{5} \right)}^{3}}{x}$

The answer is:

$\frac{4 \log{\left(\frac{x}{5} \right)}^{3}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3/x\
4*log |-|
      \5/
---------
    x

$$\frac{4 \log{\left(\frac{x}{5} \right)}^{3}}{x}$$

Simplify

The second derivative [src]

     2/x\ /       /x\\
4*log |-|*|3 - log|-||
      \5/ \       \5//
----------------------
           2          
          x

$$\frac{4 \cdot \left(3 - \log{\left(\frac{x}{5} \right)}\right) \log{\left(\frac{x}{5} \right)}^{2}}{x^{2}}$$

Simplify

The third derivative [src]

  /         /x\        2/x\\    /x\
4*|6 - 9*log|-| + 2*log |-||*log|-|
  \         \5/         \5//    \5/
-----------------------------------
                  3                
                 x

$$\frac{4 \cdot \left(2 \log{\left(\frac{x}{5} \right)}^{2} - 9 \log{\left(\frac{x}{5} \right)} + 6\right) \log{\left(\frac{x}{5} \right)}}{x^{3}}$$

Simplify

The graph

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Derivative of ln^4(x/5)

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The solution