Derivative of sin2x/sinx

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sin(2*x)
--------
 sin(x)

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

sin(2*x)/sin(x)

Detail solution

Apply the quotient rule, which is:

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

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

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. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

$- 2 \sin{\left(x \right)}$

The answer is:

$- 2 \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                /         2   \                
                                                                |    6*cos (x)|                
                                                                |5 + ---------|*cos(x)*sin(2*x)
                /         2   \                                 |        2    |                
                |    2*cos (x)|            12*cos(x)*sin(2*x)   \     sin (x) /                
-8*cos(2*x) + 6*|1 + ---------|*cos(2*x) + ------------------ - -------------------------------
                |        2    |                  sin(x)                      sin(x)            
                \     sin (x) /                                                                
-----------------------------------------------------------------------------------------------
                                             sin(x)

$$\frac{6 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(2 x \right)} - \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - 8 \cos{\left(2 x \right)} + \frac{12 \sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\sin{\left(x \right)}}$$

Simplify

The graph

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Derivative of sin2x/sinx

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The solution