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Derivative of:
Derivative of x-4
Derivative of 1/1+x^2
Derivative of e^x*sin(x)
Derivative of (x^3-4)
Identical expressions
z=ln((three x^ two + two)/(x^3+2x))^(one / four)
z equally ln((3x squared plus 2) divide by (x cubed plus 2x)) to the power of (1 divide by 4)
z equally ln((three x to the power of two plus two) divide by (x cubed plus 2x)) to the power of (one divide by four)
z=ln((3x2+2)/(x3+2x))(1/4)
z=ln3x2+2/x3+2x1/4
z=ln((3x²+2)/(x³+2x))^(1/4)
z=ln((3x to the power of 2+2)/(x to the power of 3+2x)) to the power of (1/4)
z=ln3x^2+2/x^3+2x^1/4
z=ln((3x^2+2) divide by (x^3+2x))^(1 divide by 4)
Similar expressions
z=ln((3x^2-2)/(x^3+2x))^(1/4)
z=ln((3x^2+2)/(x^3-2x))^(1/4)

The derivative of the function/ z=ln((3x^2+2)/(x^3+2x))^(1/4)

Derivative of z=ln((3x^2+2)/(x^3+2x))^(1/4)

The solution

You have entered [src]

      _______________
     /    /   2    \ 
    /     |3*x  + 2| 
   /   log|--------| 
4 /       | 3      | 
\/        \x  + 2*x/

$$\sqrt[4]{\log{\left(\frac{3 x^{2} + 2}{x^{3} + 2 x} \right)}}$$

log((3*x^2 + 2)/(x^3 + 2*x))^(1/4)

Detail solution

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

$\frac{6 x \left(x^{3} + 2 x\right) - \left(3 x^{2} + 2\right)^{2}}{4 \left(3 x^{2} + 2\right) \left(x^{3} + 2 x\right) \log{\left(\frac{3 x^{2} + 2}{x^{3} + 2 x} \right)}^{\frac{3}{4}}}$
Now simplify:

$- \frac{3 x^{4} + 4}{4 x \left(3 x^{4} + 8 x^{2} + 4\right) \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}^{\frac{3}{4}}}$

The answer is:

$- \frac{3 x^{4} + 4}{4 x \left(3 x^{4} + 8 x^{2} + 4\right) \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}^{\frac{3}{4}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           /           /        2\ /   2    \\
/ 3      \ |  6*x      \-2 - 3*x /*\3*x  + 2/|
\x  + 2*x/*|-------- + ----------------------|
           | 3                        2      |
           |x  + 2*x        / 3      \       |
           \                \x  + 2*x/       /
----------------------------------------------
                           /   2    \         
          /   2    \    3/4|3*x  + 2|         
        4*\3*x  + 2/*log   |--------|         
                           | 3      |         
                           \x  + 2*x/

$$\frac{\left(x^{3} + 2 x\right) \left(\frac{6 x}{x^{3} + 2 x} + \frac{\left(- 3 x^{2} - 2\right) \left(3 x^{2} + 2\right)}{\left(x^{3} + 2 x\right)^{2}}\right)}{4 \left(3 x^{2} + 2\right) \log{\left(\frac{3 x^{2} + 2}{x^{3} + 2 x} \right)}^{\frac{3}{4}}}$$

Simplify

The second derivative [src]

    /                              3 \                                                                           2  
    |       /       2\   /       2\  |     /              2\         /              2\          /              2\   
    |     9*\2 + 3*x /   \2 + 3*x /  |     |    /       2\ |         |    /       2\ |          |    /       2\ |   
  8*|-3 + ------------ - ------------|     |    \2 + 3*x / |       2 |    \2 + 3*x / |        2 |    \2 + 3*x / |   
    |             2                 2|   4*|6 - -----------|   24*x *|6 - -----------|     3*x *|6 - -----------|   
    |        2 + x        2 /     2\ |     |     2 /     2\|         |     2 /     2\|          |     2 /     2\|   
    \                    x *\2 + x / /     \    x *\2 + x //         \    x *\2 + x //          \    x *\2 + x //   
- ------------------------------------ + ------------------- - ----------------------- - ---------------------------
                       2                             2                         2                   2    /        2 \
                2 + 3*x                         2 + x                /       2\          /       2\     | 2 + 3*x  |
                                                                     \2 + 3*x /          \2 + 3*x / *log|----------|
                                                                                                        |  /     2\|
                                                                                                        \x*\2 + x //
--------------------------------------------------------------------------------------------------------------------
                                                        /        2 \                                                
                                                     3/4| 2 + 3*x  |                                                
                                               16*log   |----------|                                                
                                                        |  /     2\|                                                
                                                        \x*\2 + x //

$$\frac{- \frac{3 x^{2} \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)^{2}}{\left(3 x^{2} + 2\right)^{2} \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}} - \frac{24 x^{2} \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)}{\left(3 x^{2} + 2\right)^{2}} + \frac{4 \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)}{x^{2} + 2} - \frac{8 \left(-3 + \frac{9 \left(3 x^{2} + 2\right)}{x^{2} + 2} - \frac{\left(3 x^{2} + 2\right)^{3}}{x^{2} \left(x^{2} + 2\right)^{2}}\right)}{3 x^{2} + 2}}{16 \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}^{\frac{3}{4}}}$$

Simplify

The third derivative [src]

                                3        /                              3 \                                                                                 /                              4                 2\                              3                                2                                  2               /              2\ /                              3 \
         /       2\   /       2\         |       /       2\   /       2\  |         /              2\       /              2\       /              2\       |       /       2\   /       2\        /       2\ |             /              2\                /              2\                  /              2\                |    /       2\ | |       /       2\   /       2\  |
       9*\2 + 3*x /   \2 + 3*x /         |     9*\2 + 3*x /   \2 + 3*x /  |         |    /       2\ |       |    /       2\ |       |    /       2\ |       |     4*\2 + 3*x /   \2 + 3*x /     12*\2 + 3*x / |             |    /       2\ |                |    /       2\ |                  |    /       2\ |                |    \2 + 3*x / | |     9*\2 + 3*x /   \2 + 3*x /  |
  -3 + ------------ - ------------   6*x*|-3 + ------------ - ------------|       3 |    \2 + 3*x / |       |    \2 + 3*x / |       |    \2 + 3*x / |   3*x*|18 + ------------ + ------------ - --------------|           3 |    \2 + 3*x / |              3 |    \2 + 3*x / |                  |    \2 + 3*x / |            9*x*|6 - -----------|*|-3 + ------------ - ------------|
               2                 2       |             2                 2|   18*x *|6 - -----------|   3*x*|6 - -----------|   3*x*|6 - -----------|       |           2                   2     2 /     2\  |       21*x *|6 - -----------|          27*x *|6 - -----------|              9*x*|6 - -----------|                |     2 /     2\| |             2                 2|
          2 + x        2 /     2\        |        2 + x        2 /     2\ |         |     2 /     2\|       |     2 /     2\|       |     2 /     2\|       |          x          4 /     2\     x *\2 + x /  |             |     2 /     2\|                |     2 /     2\|                  |     2 /     2\|                \    x *\2 + x // |        2 + x        2 /     2\ |
                      x *\2 + x /        \                    x *\2 + x / /         \    x *\2 + x //       \    x *\2 + x //       \    x *\2 + x //       \                    x *\2 + x /                  /             \    x *\2 + x //                \    x *\2 + x //                  \    x *\2 + x //                                  \                    x *\2 + x / /
- -------------------------------- + -------------------------------------- + ----------------------- - --------------------- - --------------------- - ------------------------------------------------------- + ------------------------------- + ----------------------------- - -------------------------------------- + --------------------------------------------------------
               /     2\                                     2                                 3                         2         /     2\ /       2\                      /     2\ /       2\                                 3     /        2 \               3    /        2 \                             /        2 \                            2    /        2 \              
             x*\2 + x /                           /       2\                        /       2\                /       2\        2*\2 + x /*\2 + 3*x /                    2*\2 + x /*\2 + 3*x /                       /       2\     2| 2 + 3*x  |     /       2\     | 2 + 3*x  |      /     2\ /       2\    | 2 + 3*x  |                  /       2\     | 2 + 3*x  |              
                                                  \2 + 3*x /                        \2 + 3*x /              2*\2 + 3*x /                                                                                          64*\2 + 3*x / *log |----------|   8*\2 + 3*x / *log|----------|   16*\2 + x /*\2 + 3*x /*log|----------|                8*\2 + 3*x / *log|----------|              
                                                                                                                                                                                                                                     |  /     2\|                    |  /     2\|                             |  /     2\|                                 |  /     2\|              
                                                                                                                                                                                                                                     \x*\2 + x //                    \x*\2 + x //                             \x*\2 + x //                                 \x*\2 + x //              
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                        /        2 \                                                                                                                                                                                 
                                                                                                                                                                                     3/4| 2 + 3*x  |                                                                                                                                                                                 
                                                                                                                                                                                  log   |----------|                                                                                                                                                                                 
                                                                                                                                                                                        |  /     2\|                                                                                                                                                                                 
                                                                                                                                                                                        \x*\2 + x //

$$\frac{\frac{21 x^{3} \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)^{3}}{64 \left(3 x^{2} + 2\right)^{3} \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}^{2}} + \frac{27 x^{3} \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)^{2}}{8 \left(3 x^{2} + 2\right)^{3} \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}} + \frac{18 x^{3} \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)}{\left(3 x^{2} + 2\right)^{3}} - \frac{9 x \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)^{2}}{16 \left(x^{2} + 2\right) \left(3 x^{2} + 2\right) \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}} + \frac{9 x \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right) \left(-3 + \frac{9 \left(3 x^{2} + 2\right)}{x^{2} + 2} - \frac{\left(3 x^{2} + 2\right)^{3}}{x^{2} \left(x^{2} + 2\right)^{2}}\right)}{8 \left(3 x^{2} + 2\right)^{2} \log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}} - \frac{3 x \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)}{2 \left(3 x^{2} + 2\right)^{2}} - \frac{3 x \left(6 - \frac{\left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)}\right)}{2 \left(x^{2} + 2\right) \left(3 x^{2} + 2\right)} + \frac{6 x \left(-3 + \frac{9 \left(3 x^{2} + 2\right)}{x^{2} + 2} - \frac{\left(3 x^{2} + 2\right)^{3}}{x^{2} \left(x^{2} + 2\right)^{2}}\right)}{\left(3 x^{2} + 2\right)^{2}} - \frac{3 x \left(18 + \frac{4 \left(3 x^{2} + 2\right)}{x^{2}} - \frac{12 \left(3 x^{2} + 2\right)^{2}}{x^{2} \left(x^{2} + 2\right)} + \frac{\left(3 x^{2} + 2\right)^{4}}{x^{4} \left(x^{2} + 2\right)^{2}}\right)}{2 \left(x^{2} + 2\right) \left(3 x^{2} + 2\right)} - \frac{-3 + \frac{9 \left(3 x^{2} + 2\right)}{x^{2} + 2} - \frac{\left(3 x^{2} + 2\right)^{3}}{x^{2} \left(x^{2} + 2\right)^{2}}}{x \left(x^{2} + 2\right)}}{\log{\left(\frac{3 x^{2} + 2}{x \left(x^{2} + 2\right)} \right)}^{\frac{3}{4}}}$$

Simplify

The graph

Derivative of z=ln((3x^2+2)/(x^3+2x))^(1/4)

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of z=ln((3x^2+2)/(x^3+2x))^(1/4)

The graph:

Piecewise:

The solution