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Identical expressions
xln(one +(x)^(one / three))
xln(1 plus (x) to the power of (1 divide by 3))
xln(one plus (x) to the power of (one divide by three))
xln(1+(x)(1/3))
xln1+x1/3
xln1+x^1/3
xln(1+(x)^(1 divide by 3))
Similar expressions
xln(1-(x)^(1/3))

The derivative of the function/ xln(1+(x)^(1/3))

Derivative of xln(1+(x)^(1/3))

The solution

You have entered [src]

     /    3 ___\
x*log\1 + \/ x /

$$x \log{\left(\sqrt[3]{x} + 1 \right)}$$

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \log{\left(\sqrt[3]{x} + 1 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sqrt[3]{x} + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sqrt[3]{x} + 1\right)$ :
  1. Differentiate $\sqrt[3]{x} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Apply the power rule: $\sqrt[3]{x}$ goes to $\frac{1}{3 x^{\frac{2}{3}}}$
    The result is: $\frac{1}{3 x^{\frac{2}{3}}}$
  The result of the chain rule is:
  
  $\frac{1}{3 x^{\frac{2}{3}} \left(\sqrt[3]{x} + 1\right)}$
The result is: $\frac{\sqrt[3]{x}}{3 \left(\sqrt[3]{x} + 1\right)} + \log{\left(\sqrt[3]{x} + 1 \right)}$
Now simplify:

$\frac{\frac{\sqrt[3]{x}}{3} + \left(\sqrt[3]{x} + 1\right) \log{\left(\sqrt[3]{x} + 1 \right)}}{\sqrt[3]{x} + 1}$

The answer is:

$\frac{\frac{\sqrt[3]{x}}{3} + \left(\sqrt[3]{x} + 1\right) \log{\left(\sqrt[3]{x} + 1 \right)}}{\sqrt[3]{x} + 1}$

The graph

Plot the graph f(x)

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The first derivative [src]

    3 ___                     
    \/ x           /    3 ___\
------------- + log\1 + \/ x /
  /    3 ___\                 
3*\1 + \/ x /

$$\frac{\sqrt[3]{x}}{3 \left(\sqrt[3]{x} + 1\right)} + \log{\left(\sqrt[3]{x} + 1 \right)}$$

Simplify

The second derivative [src]

      1         4  
- --------- + -----
      3 ___   3 ___
  1 + \/ x    \/ x 
-------------------
  3 ___ /    3 ___\
9*\/ x *\1 + \/ x /

$$\frac{- \frac{1}{\sqrt[3]{x} + 1} + \frac{4}{\sqrt[3]{x}}}{9 \sqrt[3]{x} \left(\sqrt[3]{x} + 1\right)}$$

Simplify

The third derivative [src]

    /    1         2  \                                                  
  9*|--------- + -----|                                                  
    |    3 ___   3 ___|                                                  
    \1 + \/ x    \/ x /       / 5            1                 3        \
- --------------------- + 2*x*|---- + --------------- + ----------------|
            4/3               | 8/3                 2    7/3 /    3 ___\|
           x                  |x       2 /    3 ___\    x   *\1 + \/ x /|
                              \       x *\1 + \/ x /                    /
-------------------------------------------------------------------------
                                 /    3 ___\                             
                              27*\1 + \/ x /

$$\frac{2 x \left(\frac{1}{x^{2} \left(\sqrt[3]{x} + 1\right)^{2}} + \frac{3}{x^{\frac{7}{3}} \left(\sqrt[3]{x} + 1\right)} + \frac{5}{x^{\frac{8}{3}}}\right) - \frac{9 \left(\frac{1}{\sqrt[3]{x} + 1} + \frac{2}{\sqrt[3]{x}}\right)}{x^{\frac{4}{3}}}}{27 \left(\sqrt[3]{x} + 1\right)}$$

Simplify

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Derivative of xln(1+(x)^(1/3))

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The solution