Derivative of 0,5lnx^5

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   5   
log (x)
-------
   2

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

  /   5   \
d |log (x)|
--|-------|
dx\   2   /

$$\frac{d}{d x} \frac{\log{\left(x \right)}^{5}}{2}$$

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Detail solution

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

The answer is:

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

The graph

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The first derivative [src]

     4   
5*log (x)
---------
   2*x

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

Simplify

The second derivative [src]

      3                 
-5*log (x)*(-4 + log(x))
------------------------
             2          
          2*x

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

Simplify

The third derivative [src]

     2    /       2              \
5*log (x)*\6 + log (x) - 6*log(x)/
----------------------------------
                 3                
                x

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

Simplify

The graph

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Identical expressions

Derivative of 0,5lnx^5

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The solution