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(((x+ three)^ two)*(sin(x)))/((two x+ one)^2)
(((x plus 3) squared ) multiply by ( sinus of (x))) divide by ((2x plus 1) squared )
(((x plus three) to the power of two) multiply by ( sinus of (x))) divide by ((two x plus one) squared )
(((x+3)2)*(sin(x)))/((2x+1)2)
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(((x+3)²)*(sin(x)))/((2x+1)²)
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(((x+3)^2)*(sin(x))) divide by ((2x+1)^2)
Similar expressions
(((x+3)^2)*(sin(x)))/((2x-1)^2)
(((x-3)^2)*(sin(x)))/((2x+1)^2)
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The derivative of the function/ (((x+3)^2)*(sin(x)))/((2x+1)^2)

Derivative of (((x+3)^2)*(sin(x)))/((2x+1)^2)

The solution

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       2       
(x + 3) *sin(x)
---------------
            2  
   (2*x + 1)

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \left(x + 3\right)^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x + 3$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 3\right)$ :
    1. Differentiate $x + 3$ term by term:
      1. The derivative of the constant $3$ is zero.
      2. Apply the power rule: $x$ goes to $1$
      The result is: $1$
    The result of the chain rule is:
    
    $2 x + 6$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $\left(x + 3\right)^{2} \cos{\left(x \right)} + \left(2 x + 6\right) \sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x + 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 1\right)$ :
  1. Differentiate $2 x + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. 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: $2$
    The result is: $2$
  The result of the chain rule is:
  
  $8 x + 4$
Now plug in to the quotient rule:

$\frac{- \left(x + 3\right)^{2} \left(8 x + 4\right) \sin{\left(x \right)} + \left(2 x + 1\right)^{2} \left(\left(x + 3\right)^{2} \cos{\left(x \right)} + \left(2 x + 6\right) \sin{\left(x \right)}\right)}{\left(2 x + 1\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                                    2                  
(x + 3) *cos(x) + (6 + 2*x)*sin(x)   (x + 3) *(-4 - 8*x)*sin(x)
---------------------------------- + --------------------------
                     2                                4        
            (2*x + 1)                        (2*x + 1)

$$\frac{\left(- 8 x - 4\right) \left(x + 3\right)^{2} \sin{\left(x \right)}}{\left(2 x + 1\right)^{4}} + \frac{\left(x + 3\right)^{2} \cos{\left(x \right)} + \left(2 x + 6\right) \sin{\left(x \right)}}{\left(2 x + 1\right)^{2}}$$

Simplify

The second derivative [src]

                                                                                                  2       
                  2                             8*(3 + x)*(2*sin(x) + (3 + x)*cos(x))   24*(3 + x) *sin(x)
2*sin(x) - (3 + x) *sin(x) + 4*(3 + x)*cos(x) - ------------------------------------- + ------------------
                                                               1 + 2*x                               2    
                                                                                            (1 + 2*x)     
----------------------------------------------------------------------------------------------------------
                                                         2                                                
                                                (1 + 2*x)

$$\frac{- \left(x + 3\right)^{2} \sin{\left(x \right)} + \frac{24 \left(x + 3\right)^{2} \sin{\left(x \right)}}{\left(2 x + 1\right)^{2}} + 4 \left(x + 3\right) \cos{\left(x \right)} - \frac{8 \left(x + 3\right) \left(\left(x + 3\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{2 x + 1} + 2 \sin{\left(x \right)}}{\left(2 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

                                /                  2                          \                                 2                                                
                  2          12*\2*sin(x) - (3 + x) *sin(x) + 4*(3 + x)*cos(x)/                      192*(3 + x) *sin(x)   72*(3 + x)*(2*sin(x) + (3 + x)*cos(x))
6*cos(x) - (3 + x) *cos(x) - -------------------------------------------------- - 6*(3 + x)*sin(x) - ------------------- + --------------------------------------
                                                  1 + 2*x                                                          3                              2              
                                                                                                          (1 + 2*x)                      (1 + 2*x)               
-----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                     2                                                                           
                                                                            (1 + 2*x)

$$\frac{- \left(x + 3\right)^{2} \cos{\left(x \right)} - \frac{192 \left(x + 3\right)^{2} \sin{\left(x \right)}}{\left(2 x + 1\right)^{3}} - 6 \left(x + 3\right) \sin{\left(x \right)} + \frac{72 \left(x + 3\right) \left(\left(x + 3\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{\left(2 x + 1\right)^{2}} + 6 \cos{\left(x \right)} - \frac{12 \left(- \left(x + 3\right)^{2} \sin{\left(x \right)} + 4 \left(x + 3\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{2 x + 1}}{\left(2 x + 1\right)^{2}}$$

Simplify

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Derivative of (((x+3)^2)*(sin(x)))/((2x+1)^2)

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The solution