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Identical expressions
(twenty-seven (x^ two))/(two (sqrt(9x^ three)- one))
(27(x squared )) divide by (2( square root of (9x cubed ) minus 1))
(twenty minus seven (x to the power of two)) divide by (two ( square root of (9x to the power of three) minus one))
(27(x^2))/(2(√(9x^3)-1))
(27(x2))/(2(sqrt(9x3)-1))
27x2/2sqrt9x3-1
(27(x²))/(2(sqrt(9x³)-1))
(27(x to the power of 2))/(2(sqrt(9x to the power of 3)-1))
27x^2/2sqrt9x^3-1
(27(x^2)) divide by (2(sqrt(9x^3)-1))
Similar expressions
(27(x^2))/(2(sqrt(9x^3)+1))

The derivative of the function/ (27(x^2))/(2(sqrt(9x^3)-1))

Derivative of (27(x^2))/(2(sqrt(9x^3)-1))

The solution

You have entered [src]

          2      
      27*x       
-----------------
  /   ______    \
  |  /    3     |
2*\\/  9*x   - 1/

$$\frac{27 x^{2}}{2 \left(\sqrt{9 x^{3}} - 1\right)}$$

  /          2      \
d |      27*x       |
--|-----------------|
dx|  /   ______    \|
  |  |  /    3     ||
  \2*\\/  9*x   - 1//

$$\frac{d}{d x} \frac{27 x^{2}}{2 \left(\sqrt{9 x^{3}} - 1\right)}$$

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Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x^{2}$ and $g{\left(x \right)} = 6 \sqrt{x^{3}} - 2$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $6 \sqrt{x^{3}} - 2$ 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.
      1. Let $u = x^{3}$ .
      2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{3}$ :
        
        Apply the power rule: $x^{3}$ goes to $3 x^{2}$
        
        The result of the chain rule is:
        
        $\frac{3 x^{2}}{2 \sqrt{x^{3}}}$
      So, the result is: $\frac{9 x^{2}}{\sqrt{x^{3}}}$
    The result is: $\frac{9 x^{2}}{\sqrt{x^{3}}}$
  Now plug in to the quotient rule:
  
  $\frac{- \frac{9 x^{4}}{\sqrt{x^{3}}} + 2 x \left(6 \sqrt{x^{3}} - 2\right)}{\left(6 \sqrt{x^{3}} - 2\right)^{2}}$
So, the result is: $\frac{27 \left(- \frac{9 x^{4}}{\sqrt{x^{3}}} + 2 x \left(6 \sqrt{x^{3}} - 2\right)\right)}{\left(6 \sqrt{x^{3}} - 2\right)^{2}}$
Now simplify:

$\frac{81 x^{4} - 108 x \sqrt{x^{3}}}{36 x^{3} \sqrt{x^{3}} - 24 x^{3} + 4 \sqrt{x^{3}}}$

The answer is:

$\frac{81 x^{4} - 108 x \sqrt{x^{3}}}{36 x^{3} \sqrt{x^{3}} - 24 x^{3} + 4 \sqrt{x^{3}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                    ____   
                                   /  3    
             1             243*x*\/  x     
54*x*----------------- - ------------------
       /   ______    \                    2
       |  /    3     |     /   ______    \ 
     2*\\/  9*x   - 1/     |  /    3     | 
                         4*\\/  9*x   - 1/

$$54 x \frac{1}{2 \left(\sqrt{9 x^{3}} - 1\right)} - \frac{243 x \sqrt{x^{3}}}{4 \left(\sqrt{9 x^{3}} - 1\right)^{2}}$$

Simplify

The second derivative [src]

   /                          /     ____                 \\
   |                          |    /  3                  ||
   |                        2 |  \/  x          18*x     ||
   |                     9*x *|- ------- + --------------||
   |           ____           |      2               ____||
   |          /  3            |     x               /  3 ||
   |      9*\/  x             \            -1 + 3*\/  x  /|
27*|1 - -------------- + ---------------------------------|
   |              ____             /          ____\       |
   |             /  3              |         /  3 |       |
   \    -1 + 3*\/  x             8*\-1 + 3*\/  x  /       /
-----------------------------------------------------------
                                 ____                      
                                /  3                       
                       -1 + 3*\/  x

$$\frac{27 \cdot \left(\frac{9 x^{2} \cdot \left(\frac{18 x}{3 \sqrt{x^{3}} - 1} - \frac{\sqrt{x^{3}}}{x^{2}}\right)}{8 \cdot \left(3 \sqrt{x^{3}} - 1\right)} + 1 - \frac{9 \sqrt{x^{3}}}{3 \sqrt{x^{3}} - 1}\right)}{3 \sqrt{x^{3}} - 1}$$

Simplify

The third derivative [src]

    /   /                    ____             ____   \         ____        /     ____                 \\
    |   |                   /  3             /  3    |        /  3         |    /  3                  ||
    | 2 |      54         \/  x        486*\/  x     |   24*\/  x          |  \/  x          18*x     ||
243*|x *|-------------- + ------- - -----------------| - ---------- + 12*x*|- ------- + --------------||
    |   |          ____       3                     2|       x             |      2               ____||
    |   |         /  3       x      /          ____\ |                     |     x               /  3 ||
    |   |-1 + 3*\/  x               |         /  3 | |                     \            -1 + 3*\/  x  /|
    \   \                           \-1 + 3*\/  x  / /                                                 /
--------------------------------------------------------------------------------------------------------
                                                             2                                          
                                             /          ____\                                           
                                             |         /  3 |                                           
                                          16*\-1 + 3*\/  x  /

$$\frac{243 \left(x^{2} \cdot \left(\frac{54}{3 \sqrt{x^{3}} - 1} - \frac{486 \sqrt{x^{3}}}{\left(3 \sqrt{x^{3}} - 1\right)^{2}} + \frac{\sqrt{x^{3}}}{x^{3}}\right) + 12 x \left(\frac{18 x}{3 \sqrt{x^{3}} - 1} - \frac{\sqrt{x^{3}}}{x^{2}}\right) - \frac{24 \sqrt{x^{3}}}{x}\right)}{16 \left(3 \sqrt{x^{3}} - 1\right)^{2}}$$

Simplify

The graph

Derivative of (27(x^2))/(2(sqrt(9x^3)-1))

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How to use it?

Identical expressions

Similar expressions

Derivative of (27(x^2))/(2(sqrt(9x^3)-1))

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Piecewise:

The solution