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Identical expressions
(x^ five ^(one / six)- three *x^(one / three)+ two)/(x^ three)^(one / four)
(x to the power of 5 to the power of (1 divide by 6) minus 3 multiply by x to the power of (1 divide by 3) plus 2) divide by (x cubed ) to the power of (1 divide by 4)
(x to the power of five to the power of (one divide by six) minus three multiply by x to the power of (one divide by three) plus two) divide by (x to the power of three) to the power of (one divide by four)
(x5(1/6)-3*x(1/3)+2)/(x3)(1/4)
x51/6-3*x1/3+2/x31/4
(x⁵^(1/6)-3*x^(1/3)+2)/(x³)^(1/4)
(x to the power of 5 to the power of (1/6)-3*x to the power of (1/3)+2)/(x to the power of 3) to the power of (1/4)
(x^5^(1/6)-3x^(1/3)+2)/(x^3)^(1/4)
(x5(1/6)-3x(1/3)+2)/(x3)(1/4)
x51/6-3x1/3+2/x31/4
x^5^1/6-3x^1/3+2/x^3^1/4
(x^5^(1 divide by 6)-3*x^(1 divide by 3)+2) divide by (x^3)^(1 divide by 4)
Similar expressions
(x^5^(1/6)-3*x^(1/3)-2)/(x^3)^(1/4)
(x^5^(1/6)+3*x^(1/3)+2)/(x^3)^(1/4)

The derivative of the function/ (x^5^(1/6)-3*x^(1/3)+2)/(x^3)^(1/4)

Derivative of (x^5^(1/6)-3*x^(1/3)+2)/(x^3)^(1/4)

The solution

You have entered [src]

 6 ___              
 \/ 5      3 ___    
x      - 3*\/ x  + 2
--------------------
         ____       
      4 /  3        
      \/  x

$$\frac{- 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2}{\sqrt[4]{x^{3}}}$$

  / 6 ___              \
  | \/ 5      3 ___    |
d |x      - 3*\/ x  + 2|
--|--------------------|
dx|         ____       |
  |      4 /  3        |
  \      \/  x         /

$$\frac{d}{d x} \frac{- 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2}{\sqrt[4]{x^{3}}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = - 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2$ and $g{\left(x \right)} = \sqrt[4]{x^{3}}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. Apply the power rule: $x^{\sqrt[6]{5}}$ goes to $\frac{\sqrt[6]{5} x^{\sqrt[6]{5}}}{x}$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\sqrt[3]{x}$ goes to $\frac{1}{3 x^{\frac{2}{3}}}$
    So, the result is: $- \frac{1}{x^{\frac{2}{3}}}$
  The result is: $\frac{\sqrt[6]{5} x^{\sqrt[6]{5}}}{x} - \frac{1}{x^{\frac{2}{3}}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{3}$ .
2. Apply the power rule: $\sqrt[4]{u}$ goes to $\frac{1}{4 u^{\frac{3}{4}}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{3}$ :
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  The result of the chain rule is:
  
  $\frac{3 x^{2}}{4 \left(x^{3}\right)^{\frac{3}{4}}}$
Now plug in to the quotient rule:

$\frac{- \frac{3 x^{2} \left(- 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2\right)}{4 \left(x^{3}\right)^{\frac{3}{4}}} + \left(\frac{\sqrt[6]{5} x^{\sqrt[6]{5}}}{x} - \frac{1}{x^{\frac{2}{3}}}\right) \sqrt[4]{x^{3}}}{\sqrt{x^{3}}}$
Now simplify:

$\frac{5 x^{\frac{7}{3}} - 6 x^{2} - 3 x^{\sqrt[6]{5} + 2} + 4 \cdot \sqrt[6]{5} x^{\sqrt[6]{5} + 2}}{4 \left(x^{3}\right)^{\frac{5}{4}}}$

The answer is:

$\frac{5 x^{\frac{7}{3}} - 6 x^{2} - 3 x^{\sqrt[6]{5} + 2} + 4 \cdot \sqrt[6]{5} x^{\sqrt[6]{5} + 2}}{4 \left(x^{3}\right)^{\frac{5}{4}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                6 ___                           
         6 ___  \/ 5                            
   1     \/ 5 *x                                
- ---- + ------------     / 6 ___              \
   2/3        x           | \/ 5      3 ___    |
  x                     3*\x      - 3*\/ x  + 2/
--------------------- - ------------------------
          ____                       ____       
       4 /  3                     4 /  3        
       \/  x                  4*x*\/  x

$$\frac{\frac{\sqrt[6]{5} x^{\sqrt[6]{5}}}{x} - \frac{1}{x^{\frac{2}{3}}}}{\sqrt[4]{x^{3}}} - \frac{3 \left(- 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2\right)}{4 x \sqrt[4]{x^{3}}}$$

Simplify

The second derivative [src]

           /                6 ___\                                                          
           |         6 ___  \/ 5 |                                                          
           |   1     \/ 5 *x     |                                                          
         3*|- ---- + ------------|      /     6 ___          \          6 ___          6 ___
           |   2/3        x      |      |     \/ 5      3 ___|   3 ___  \/ 5    6 ___  \/ 5 
  2        \  x                  /   21*\2 + x      - 3*\/ x /   \/ 5 *x        \/ 5 *x     
------ - ------------------------- + ------------------------- + ------------ - ------------
   5/3              2*x                            2                   2              2     
3*x                                            16*x                   x              x      
--------------------------------------------------------------------------------------------
                                             ____                                           
                                          4 /  3                                            
                                          \/  x

$$\frac{- \frac{3 \cdot \left(\frac{\sqrt[6]{5} x^{\sqrt[6]{5}}}{x} - \frac{1}{x^{\frac{2}{3}}}\right)}{2 x} - \frac{\sqrt[6]{5} x^{\sqrt[6]{5}}}{x^{2}} + \frac{\sqrt[3]{5} x^{\sqrt[6]{5}}}{x^{2}} + \frac{21 \left(- 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2\right)}{16 x^{2}} + \frac{2}{3 x^{\frac{5}{3}}}}{\sqrt[4]{x^{3}}}$$

Simplify

The third derivative [src]

                                          /                6 ___            6 ___\      /                6 ___\                                                 
                                          |         6 ___  \/ 5      3 ___  \/ 5 |      |         6 ___  \/ 5 |                                                 
                                          | 2     3*\/ 5 *x        3*\/ 5 *x     |      |   1     \/ 5 *x     |                                                 
               /     6 ___          \   3*|---- - -------------- + --------------|   63*|- ---- + ------------|          6 ___            6 ___            6 ___
               |     \/ 5      3 ___|     | 5/3          2                2      |      |   2/3        x      |     ___  \/ 5      3 ___  \/ 5      6 ___  \/ 5 
    10     231*\2 + x      - 3*\/ x /     \x            x                x       /      \  x                  /   \/ 5 *x        3*\/ 5 *x        2*\/ 5 *x     
- ------ - -------------------------- - ------------------------------------------ + -------------------------- + ------------ - -------------- + --------------
     8/3                 3                                 4*x                                     2                    3               3                3      
  9*x                64*x                                                                      16*x                    x               x                x       
----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               ____                                                                             
                                                                            4 /  3                                                                              
                                                                            \/  x

$$\frac{- \frac{3 \left(- \frac{3 \cdot \sqrt[6]{5} x^{\sqrt[6]{5}}}{x^{2}} + \frac{3 \cdot \sqrt[3]{5} x^{\sqrt[6]{5}}}{x^{2}} + \frac{2}{x^{\frac{5}{3}}}\right)}{4 x} + \frac{63 \cdot \left(\frac{\sqrt[6]{5} x^{\sqrt[6]{5}}}{x} - \frac{1}{x^{\frac{2}{3}}}\right)}{16 x^{2}} - \frac{3 \cdot \sqrt[3]{5} x^{\sqrt[6]{5}}}{x^{3}} + \frac{\sqrt{5} x^{\sqrt[6]{5}}}{x^{3}} + \frac{2 \cdot \sqrt[6]{5} x^{\sqrt[6]{5}}}{x^{3}} - \frac{231 \left(- 3 \sqrt[3]{x} + x^{\sqrt[6]{5}} + 2\right)}{64 x^{3}} - \frac{10}{9 x^{\frac{8}{3}}}}{\sqrt[4]{x^{3}}}$$

Simplify

The graph

Derivative of (x^5^(1/6)-3*x^(1/3)+2)/(x^3)^(1/4)

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Identical expressions

Similar expressions

Derivative of (x^5^(1/6)-3*x^(1/3)+2)/(x^3)^(1/4)

The graph:

Piecewise:

The solution