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sin(x)/(two x+ one)^2
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Similar expressions
sin(x)/(2x-1)^2
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The derivative of the function/ sin(x)/(2x+1)^2

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

          192*sin(x)   12*sin(x)   72*cos(x) 
-cos(x) - ---------- + --------- + ----------
                   3    1 + 2*x             2
          (1 + 2*x)                (1 + 2*x) 
---------------------------------------------
                           2                 
                  (1 + 2*x)

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

Simplify

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

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The solution