Derivative of ln^5(x)

The solution

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   5   
log (x)

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

log(x)^5

Detail solution

Let $u = \log{\left(x \right)}$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

$$\frac{10 \left(\log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 6\right) \log{\left(x \right)}^{2}}{x^{3}}$$

Simplify

The graph

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Identical expressions

Derivative of ln^5(x)

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The solution