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Identical expressions
√³sin²*5x/(x³+ one)
√³ sinus of ² multiply by 5x divide by (x³ plus 1)
√³ sinus of ² multiply by 5x divide by (x³ plus one)
√³sin²5x/(x³+1)
√³sin²5x/x³+1
√³sin²*5x divide by (x³+1)
Similar expressions
√³sin²*5x/(x³-1)

The derivative of the function/ √³sin²*5x/(x³+1)

Derivative of √³sin²*5x/(x³+1)

The solution

You have entered [src]

     3          
  ___     2     
\/ x  *sin (5)*x
----------------
      3         
     x  + 1

$$\frac{x \left(\sqrt{x}\right)^{3} \sin^{2}{\left(5 \right)}}{x^{3} + 1}$$

(((sqrt(x))^3*sin(5)^2)*x)/(x^3 + 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{\frac{5}{2}} \sin^{2}{\left(5 \right)}$ and $g{\left(x \right)} = x^{3} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{\frac{5}{2}}$ goes to $\frac{5 x^{\frac{3}{2}}}{2}$
  So, the result is: $\frac{5 x^{\frac{3}{2}} \sin^{2}{\left(5 \right)}}{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{3} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  The result is: $3 x^{2}$
Now plug in to the quotient rule:

$\frac{- 3 x^{\frac{9}{2}} \sin^{2}{\left(5 \right)} + \frac{5 x^{\frac{3}{2}} \left(x^{3} + 1\right) \sin^{2}{\left(5 \right)}}{2}}{\left(x^{3} + 1\right)^{2}}$
Now simplify:

$\frac{x^{\frac{3}{2}} \left(5 - x^{3}\right) \sin^{2}{\left(5 \right)}}{2 \left(x^{3} + 1\right)^{2}}$

The answer is:

$\frac{x^{\frac{3}{2}} \left(5 - x^{3}\right) \sin^{2}{\left(5 \right)}}{2 \left(x^{3} + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3              3/2    2                       
  ___     2      3*x   *sin (5)                    
\/ x  *sin (5) + --------------      3  3/2    2   
                       2          3*x *x   *sin (5)
------------------------------- - -----------------
              3                               2    
             x  + 1                   / 3    \     
                                      \x  + 1/

$$- \frac{3 x^{\frac{3}{2}} x^{3} \sin^{2}{\left(5 \right)}}{\left(x^{3} + 1\right)^{2}} + \frac{\left(\sqrt{x}\right)^{3} \sin^{2}{\left(5 \right)} + \frac{3 x^{\frac{3}{2}} \sin^{2}{\left(5 \right)}}{2}}{x^{3} + 1}$$

Simplify

The second derivative [src]

                /                  /         3 \\
                |                3 |      3*x  ||
                |             2*x *|-1 + ------||
                |        3         |          3||
    ___    2    |5    5*x          \     1 + x /|
3*\/ x *sin (5)*|- - ------ + ------------------|
                |4        3              3      |
                \    1 + x          1 + x       /
-------------------------------------------------
                           3                     
                      1 + x

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

Simplify

The third derivative [src]

          /                              /        3          6  \                        \
          |                          5/2 |    18*x       27*x   |           /         3 \|
          |                       2*x   *|1 - ------ + ---------|       5/2 |      3*x  ||
          |                              |         3           2|   15*x   *|-1 + ------||
          |               5/2            |    1 + x    /     3\ |           |          3||
     2    |   5       45*x               \             \1 + x / /           \     1 + x /|
3*sin (5)*|------- - ---------- - ------------------------------- + ---------------------|
          |    ___     /     3\                     3                            3       |
          \8*\/ x    4*\1 + x /                1 + x                        1 + x        /
------------------------------------------------------------------------------------------
                                               3                                          
                                          1 + x

$$\frac{3 \left(\frac{15 x^{\frac{5}{2}} \left(\frac{3 x^{3}}{x^{3} + 1} - 1\right)}{x^{3} + 1} - \frac{2 x^{\frac{5}{2}} \left(\frac{27 x^{6}}{\left(x^{3} + 1\right)^{2}} - \frac{18 x^{3}}{x^{3} + 1} + 1\right)}{x^{3} + 1} - \frac{45 x^{\frac{5}{2}}}{4 \left(x^{3} + 1\right)} + \frac{5}{8 \sqrt{x}}\right) \sin^{2}{\left(5 \right)}}{x^{3} + 1}$$

Simplify

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Derivative of √³sin²*5x/(x³+1)

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The solution