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Similar expressions
y=sin(3x/(x^3-1))

The derivative of the function/ y=sin(3x/(x^3+1))

Derivative of y=sin(3x/(x^3+1))

The solution

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   / 3*x  \
sin|------|
   | 3    |
   \x  + 1/

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

sin((3*x)/(x^3 + 1))

Detail solution

Let $u = \frac{3 x}{x^{3} + 1}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{3 x}{x^{3} + 1}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 3 x$ 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: $x$ goes to $1$
    So, the result is: $3$
  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:
  
  $\frac{3 - 6 x^{3}}{\left(x^{3} + 1\right)^{2}}$
The result of the chain rule is:

$\frac{\left(3 - 6 x^{3}\right) \cos{\left(\frac{3 x}{x^{3} + 1} \right)}}{\left(x^{3} + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/               3  \            
|  3         9*x   |    / 3*x  \
|------ - ---------|*cos|------|
| 3               2|    | 3    |
|x  + 1   / 3    \ |    \x  + 1/
\         \x  + 1/ /

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

Simplify

The second derivative [src]

  /               2                                             \
  |  /         3 \                     /         3 \            |
  |  |      3*x  |     / 3*x  \      2 |      3*x  |    / 3*x  \|
9*|- |-1 + ------| *sin|------| + 2*x *|-2 + ------|*cos|------||
  |  |          3|     |     3|        |          3|    |     3||
  \  \     1 + x /     \1 + x /        \     1 + x /    \1 + x //
-----------------------------------------------------------------
                                    2                            
                            /     3\                             
                            \1 + x /

$$\frac{9 \left(2 x^{2} \left(\frac{3 x^{3}}{x^{3} + 1} - 2\right) \cos{\left(\frac{3 x}{x^{3} + 1} \right)} - \left(\frac{3 x^{3}}{x^{3} + 1} - 1\right)^{2} \sin{\left(\frac{3 x}{x^{3} + 1} \right)}\right)}{\left(x^{3} + 1\right)^{2}}$$

Simplify

The third derivative [src]

  /                                                            3                                                            \
  |                                               /         3 \                      /         3 \ /         3 \            |
  |                                               |      3*x  |     / 3*x  \       2 |      3*x  | |      3*x  |    / 3*x  \|
  |                                             3*|-1 + ------| *cos|------|   18*x *|-1 + ------|*|-2 + ------|*sin|------||
  |      /        3          6  \                 |          3|     |     3|         |          3| |          3|    |     3||
  |      |    27*x       27*x   |    / 3*x  \     \     1 + x /     \1 + x /         \     1 + x / \     1 + x /    \1 + x /|
9*|- 2*x*|4 - ------ + ---------|*cos|------| + ---------------------------- + ---------------------------------------------|
  |      |         3           2|    |     3|                   3                                       3                   |
  |      |    1 + x    /     3\ |    \1 + x /              1 + x                                   1 + x                    |
  \      \             \1 + x / /                                                                                           /
-----------------------------------------------------------------------------------------------------------------------------
                                                                  2                                                          
                                                          /     3\                                                           
                                                          \1 + x /

$$\frac{9 \left(\frac{18 x^{2} \left(\frac{3 x^{3}}{x^{3} + 1} - 2\right) \left(\frac{3 x^{3}}{x^{3} + 1} - 1\right) \sin{\left(\frac{3 x}{x^{3} + 1} \right)}}{x^{3} + 1} - 2 x \left(\frac{27 x^{6}}{\left(x^{3} + 1\right)^{2}} - \frac{27 x^{3}}{x^{3} + 1} + 4\right) \cos{\left(\frac{3 x}{x^{3} + 1} \right)} + \frac{3 \left(\frac{3 x^{3}}{x^{3} + 1} - 1\right)^{3} \cos{\left(\frac{3 x}{x^{3} + 1} \right)}}{x^{3} + 1}\right)}{\left(x^{3} + 1\right)^{2}}$$

Simplify

The graph

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Derivative of y=sin(3x/(x^3+1))

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The solution