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

Derivative of √³sin²*5x/(x³+1)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
     3          
  ___     2     
\/ x  *sin (5)*x
----------------
      3         
     x  + 1
x(x)3sin⁡2(5)x3+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

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x52sin⁡2(5)f{\left(x \right)} = x^{\frac{5}{2}} \sin^{2}{\left(5 \right)} and g(x)=x3+1g{\left(x \right)} = x^{3} + 1.

    To find ddxf(x)\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: x52x^{\frac{5}{2}} goes to 5x322\frac{5 x^{\frac{3}{2}}}{2}

      So, the result is: 5x32sin⁡2(5)2\frac{5 x^{\frac{3}{2}} \sin^{2}{\left(5 \right)}}{2}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x3+1x^{3} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      The result is: 3x23 x^{2}

    Now plug in to the quotient rule:

    −3x92sin⁡2(5)+5x32(x3+1)sin⁡2(5)2(x3+1)2\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}}

  2. Now simplify:

    x32(5−x3)sin⁡2(5)2(x3+1)2\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:

x32(5−x3)sin⁡2(5)2(x3+1)2\frac{x^{\frac{3}{2}} \left(5 - x^{3}\right) \sin^{2}{\left(5 \right)}}{2 \left(x^{3} + 1\right)^{2}}

The graph
02468-8-6-4-2-10101.0-1.0
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/
−3x32x3sin⁡2(5)(x3+1)2+(x)3sin⁡2(5)+3x32sin⁡2(5)2x3+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
3x(2x3(3x3x3+1−1)x3+1−5x3x3+1+54)sin⁡2(5)x3+1\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
3(15x52(3x3x3+1−1)x3+1−2x52(27x6(x3+1)2−18x3x3+1+1)x3+1−45x524(x3+1)+58x)sin⁡2(5)x3+1\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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