Derivative of lnx^40

The solution

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   40   
log  (x)

$$\log{\left(x \right)}^{40}$$

log(x)^40

Detail solution

Let $u = \log{\left(x \right)}$ .
Apply the power rule: $u^{40}$ goes to $40 u^{39}$
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:

$\frac{40 \log{\left(x \right)}^{39}}{x}$

The answer is:

$\frac{40 \log{\left(x \right)}^{39}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      39   
40*log  (x)
-----------
     x

$$\frac{40 \log{\left(x \right)}^{39}}{x}$$

Simplify

The second derivative [src]

      38                 
40*log  (x)*(39 - log(x))
-------------------------
             2           
            x

$$\frac{40 \left(39 - \log{\left(x \right)}\right) \log{\left(x \right)}^{38}}{x^{2}}$$

Simplify

The third derivative [src]

      37    /                         2   \
40*log  (x)*\1482 - 117*log(x) + 2*log (x)/
-------------------------------------------
                      3                    
                     x

$$\frac{40 \left(2 \log{\left(x \right)}^{2} - 117 \log{\left(x \right)} + 1482\right) \log{\left(x \right)}^{37}}{x^{3}}$$

Simplify

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Identical expressions

Derivative of lnx^40

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The solution