Derivative of ln^7(x)

The solution

You have entered [src]

   7   
log (x)

$$\log{\left(x \right)}^{7}$$

log(x)^7

Detail solution

Let $u = \log{\left(x \right)}$ .
Apply the power rule: $u^{7}$ goes to $7 u^{6}$
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:

$\frac{7 \log{\left(x \right)}^{6}}{x}$

The answer is:

$\frac{7 \log{\left(x \right)}^{6}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     6   
7*log (x)
---------
    x

$$\frac{7 \log{\left(x \right)}^{6}}{x}$$

Simplify

The second derivative [src]

     5                
7*log (x)*(6 - log(x))
----------------------
           2          
          x

$$\frac{7 \left(6 - \log{\left(x \right)}\right) \log{\left(x \right)}^{5}}{x^{2}}$$

Simplify

The third derivative [src]

      4    /        2              \
14*log (x)*\15 + log (x) - 9*log(x)/
------------------------------------
                  3                 
                 x

$$\frac{14 \left(\log{\left(x \right)}^{2} - 9 \log{\left(x \right)} + 15\right) \log{\left(x \right)}^{4}}{x^{3}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of ln^7(x)

The graph:

Piecewise:

The solution