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Similar expressions
3sinx/(3x^2-3sinx)

The derivative of the function/ 3sinx/(3x^2+3sinx)

Derivative of 3sinx/(3x^2+3sinx)

The solution

You have entered [src]

    3*sin(x)   
---------------
   2           
3*x  + 3*sin(x)

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

(3*sin(x))/(3*x^2 + 3*sin(x))

Detail solution

Apply the quotient rule, which is:

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

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

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. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $3 \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x^{2} + 3 \sin{\left(x \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $6 x$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    So, the result is: $3 \cos{\left(x \right)}$
  The result is: $6 x + 3 \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    3*cos(x)      3*(-6*x - 3*cos(x))*sin(x)
--------------- + --------------------------
   2                                   2    
3*x  + 3*sin(x)       /   2           \     
                      \3*x  + 3*sin(x)/

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

Simplify

The second derivative [src]

          /                     2         \                                 
          |     2*(2*x + cos(x))          |                                 
          |-2 + ----------------- + sin(x)|*sin(x)                          
          |         2                     |                                 
          \        x  + sin(x)            /          2*(2*x + cos(x))*cos(x)
-sin(x) + ---------------------------------------- - -----------------------
                         2                                  2               
                        x  + sin(x)                        x  + sin(x)      
----------------------------------------------------------------------------
                                 2                                          
                                x  + sin(x)

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

Simplify

The third derivative [src]

          /                          3                                 \                                                                              
          |          6*(2*x + cos(x))    6*(-2 + sin(x))*(2*x + cos(x))|                                      /                     2         \       
          |-cos(x) + ----------------- + ------------------------------|*sin(x)                               |     2*(2*x + cos(x))          |       
          |                         2              2                   |                                    3*|-2 + ----------------- + sin(x)|*cos(x)
          |            / 2         \              x  + sin(x)          |                                      |         2                     |       
          \            \x  + sin(x)/                                   /          3*(2*x + cos(x))*sin(x)     \        x  + sin(x)            /       
-cos(x) - --------------------------------------------------------------------- + ----------------------- + ------------------------------------------
                                        2                                                2                                  2                         
                                       x  + sin(x)                                      x  + sin(x)                        x  + sin(x)                
------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                      2                                                                               
                                                                     x  + sin(x)

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

Simplify

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Derivative of 3sinx/(3x^2+3sinx)

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The solution