Derivative of (x^⅓)ln²x

You have entered [src]

3 ___    2   
\/ x *log (x)

$$\sqrt[3]{x} \log{\left(x \right)}^{2}$$

x^(1/3)*log(x)^2

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sqrt[3]{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt[3]{x}$ goes to $\frac{1}{3 x^{\frac{2}{3}}}$
$g{\left(x \right)} = \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
The result is: $\frac{\log{\left(x \right)}^{2}}{3 x^{\frac{2}{3}}} + \frac{2 \log{\left(x \right)}}{x^{\frac{2}{3}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                2   \
  |    log(x)   log (x)|
2*|1 - ------ - -------|
  \      3         9   /
------------------------
           5/3          
          x

$$\frac{2 \left(- \frac{\log{\left(x \right)}^{2}}{9} - \frac{\log{\left(x \right)}}{3} + 1\right)}{x^{\frac{5}{3}}}$$

Simplify

The third derivative [src]

  /                   2   \
  |     log(x)   5*log (x)|
2*|-2 + ------ + ---------|
  \       3          27   /
---------------------------
             8/3           
            x

$$\frac{2 \left(\frac{5 \log{\left(x \right)}^{2}}{27} + \frac{\log{\left(x \right)}}{3} - 2\right)}{x^{\frac{8}{3}}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of (x^⅓)ln²x

The graph:

Piecewise:

The solution