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Identical expressions
x*sinx+(3x/(x+ two))
x multiply by sinus of x plus (3x divide by (x plus 2))
x multiply by sinus of x plus (3x divide by (x plus two))
xsinx+(3x/(x+2))
xsinx+3x/x+2
x*sinx+(3x divide by (x+2))
Similar expressions
x*sinx+(3x/(x-2))
x*sinx-(3x/(x+2))

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

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

The solution

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

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

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

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

               18                   18*x  
-3*sin(x) + -------- - x*cos(x) - --------
                   3                     4
            (2 + x)               (2 + x)

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

Simplify

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

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The solution