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Derivative of cbrt(x^5)*ln(x)

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   ____       
3 /  5        
\/  x  *log(x)

$$\sqrt[3]{x^{5}} \log{\left(x \right)}$$

  /   ____       \
d |3 /  5        |
--\\/  x  *log(x)/
dx

$$\frac{d}{d x} \sqrt[3]{x^{5}} \log{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt[3]{x^{5}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{5}$ .
2. Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{5}$ :
  1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
  The result of the chain rule is:
  
  $\frac{5 x^{4}}{3 \left(x^{5}\right)^{\frac{2}{3}}}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
The result is: $\frac{5 x^{4} \log{\left(x \right)}}{3 \left(x^{5}\right)^{\frac{2}{3}}} + \frac{\sqrt[3]{x^{5}}}{x}$
Now simplify:

$\frac{x^{4} \cdot \left(5 \log{\left(x \right)} + 3\right)}{3 \left(x^{5}\right)^{\frac{2}{3}}}$

The answer is:

$\frac{x^{4} \cdot \left(5 \log{\left(x \right)} + 3\right)}{3 \left(x^{5}\right)^{\frac{2}{3}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   ____        ____       
3 /  5      3 /  5        
\/  x     5*\/  x  *log(x)
------- + ----------------
   x            3*x

$$\frac{5 \sqrt[3]{x^{5}} \log{\left(x \right)}}{3 x} + \frac{\sqrt[3]{x^{5}}}{x}$$

Simplify

The second derivative [src]

   ____                 
3 /  5                  
\/  x  *(21 + 10*log(x))
------------------------
             2          
          9*x

$$\frac{\left(10 \log{\left(x \right)} + 21\right) \sqrt[3]{x^{5}}}{9 x^{2}}$$

Simplify

The third derivative [src]

   ____                
3 /  5                 
\/  x  *(9 - 10*log(x))
-----------------------
             3         
         27*x

$$\frac{\left(9 - 10 \log{\left(x \right)}\right) \sqrt[3]{x^{5}}}{27 x^{3}}$$

Simplify

The graph