Mister Exam

Lang:

Derivative of ln^3(x)

You have entered [src]

   3   
log (x)

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

d /   3   \
--\log (x)/
dx

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

Transform

Detail solution

Let $u = \log{\left(x \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2              \
6*\1 + log (x) - 3*log(x)/
--------------------------
             3            
            x

$$\frac{6 \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x^{3}}$$

Simplify

The graph