Derivative of y=(sin2x)/(2x+1)

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
  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: $2$
  The result of the chain rule is:
  
  $2 \cos{\left(2 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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: $2$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /            2*cos(2*x)   2*sin(2*x)\
4*|-sin(2*x) - ---------- + ----------|
  |             1 + 2*x              2|
  \                         (1 + 2*x) /
---------------------------------------
                1 + 2*x

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

Simplify

The third derivative [src]

  /            6*sin(2*x)   3*sin(2*x)   6*cos(2*x)\
8*|-cos(2*x) - ---------- + ---------- + ----------|
  |                     3    1 + 2*x              2|
  \            (1 + 2*x)                 (1 + 2*x) /
----------------------------------------------------
                      1 + 2*x

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

Simplify

The graph

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Derivative of y=(sin2x)/(2x+1)

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The solution