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Identical expressions
sin(2x)/(sin(x+ zero . eighty-one))
sinus of (2x) divide by ( sinus of (x plus 0.81))
sinus of (2x) divide by ( sinus of (x plus zero . eighty minus one))
sin2x/sinx+0.81
sin(2x) divide by (sin(x+0.81))
Similar expressions
sin(2x)/(sin(x-0.81))

The derivative of the function/ sin(2x)/(sin(x+0.81))

Derivative of sin(2x)/(sin(x+0.81))

The solution

You have entered [src]

  sin(2*x)  
------------
   /     81\
sin|x + ---|
   \    100/

$$\frac{\sin{\left(2 x \right)}}{\sin{\left(x + \frac{81}{100} \right)}}$$

sin(2*x)/sin(x + 81/100)

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 $g{\left(x \right)} = \sin{\left(x + \frac{81}{100} \right)}$ .

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. Let $u = x + \frac{81}{100}$ .
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 $\frac{d}{d x} \left(x + \frac{81}{100}\right)$ :
  1. Differentiate $x + \frac{81}{100}$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $\frac{81}{100}$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\cos{\left(x + \frac{81}{100} \right)}$
Now plug in to the quotient rule:

$\frac{- \sin{\left(2 x \right)} \cos{\left(x + \frac{81}{100} \right)} + 2 \sin{\left(x + \frac{81}{100} \right)} \cos{\left(2 x \right)}}{\sin^{2}{\left(x + \frac{81}{100} \right)}}$
Now simplify:

$\frac{- 3 \sin{\left(x - \frac{81}{100} \right)} + \sin{\left(3 x + \frac{81}{100} \right)}}{2 \sin^{2}{\left(x + \frac{81}{100} \right)}}$

The answer is:

$\frac{- 3 \sin{\left(x - \frac{81}{100} \right)} + \sin{\left(3 x + \frac{81}{100} \right)}}{2 \sin^{2}{\left(x + \frac{81}{100} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                  /     81\         
               cos|x + ---|*sin(2*x)
 2*cos(2*x)       \    100/         
------------ - ---------------------
   /     81\          2/     81\    
sin|x + ---|       sin |x + ---|    
   \    100/           \    100/

$$- \frac{\sin{\left(2 x \right)} \cos{\left(x + \frac{81}{100} \right)}}{\sin^{2}{\left(x + \frac{81}{100} \right)}} + \frac{2 \cos{\left(2 x \right)}}{\sin{\left(x + \frac{81}{100} \right)}}$$

Simplify

The second derivative [src]

              /         2/ 81    \\                          / 81    \
              |    2*cos |--- + x||            4*cos(2*x)*cos|--- + x|
              |          \100    /|                          \100    /
-4*sin(2*x) + |1 + ---------------|*sin(2*x) - -----------------------
              |        2/ 81    \ |                     / 81    \     
              |     sin |--- + x| |                  sin|--- + x|     
              \         \100    / /                     \100    /     
----------------------------------------------------------------------
                                / 81    \                             
                             sin|--- + x|                             
                                \100    /

$$\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{81}{100} \right)}}{\sin^{2}{\left(x + \frac{81}{100} \right)}}\right) \sin{\left(2 x \right)} - 4 \sin{\left(2 x \right)} - \frac{4 \cos{\left(2 x \right)} \cos{\left(x + \frac{81}{100} \right)}}{\sin{\left(x + \frac{81}{100} \right)}}}{\sin{\left(x + \frac{81}{100} \right)}}$$

Simplify

The third derivative [src]

                                                                            /         2/ 81    \\                      
                                                                            |    6*cos |--- + x||                      
                                                                            |          \100    /|    / 81    \         
                                                                            |5 + ---------------|*cos|--- + x|*sin(2*x)
                /         2/ 81    \\                  / 81    \            |        2/ 81    \ |    \100    /         
                |    2*cos |--- + x||            12*cos|--- + x|*sin(2*x)   |     sin |--- + x| |                      
                |          \100    /|                  \100    /            \         \100    / /                      
-8*cos(2*x) + 6*|1 + ---------------|*cos(2*x) + ------------------------ - -------------------------------------------
                |        2/ 81    \ |                     / 81    \                            / 81    \               
                |     sin |--- + x| |                  sin|--- + x|                         sin|--- + x|               
                \         \100    / /                     \100    /                            \100    /               
-----------------------------------------------------------------------------------------------------------------------
                                                         / 81    \                                                     
                                                      sin|--- + x|                                                     
                                                         \100    /

$$\frac{6 \left(1 + \frac{2 \cos^{2}{\left(x + \frac{81}{100} \right)}}{\sin^{2}{\left(x + \frac{81}{100} \right)}}\right) \cos{\left(2 x \right)} - \frac{\left(5 + \frac{6 \cos^{2}{\left(x + \frac{81}{100} \right)}}{\sin^{2}{\left(x + \frac{81}{100} \right)}}\right) \sin{\left(2 x \right)} \cos{\left(x + \frac{81}{100} \right)}}{\sin{\left(x + \frac{81}{100} \right)}} + \frac{12 \sin{\left(2 x \right)} \cos{\left(x + \frac{81}{100} \right)}}{\sin{\left(x + \frac{81}{100} \right)}} - 8 \cos{\left(2 x \right)}}{\sin{\left(x + \frac{81}{100} \right)}}$$

Simplify

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Derivative of sin(2x)/(sin(x+0.81))

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