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sinx/(eight *cos(x)*sin(x))
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sinx/(8*cosx*sinx)

The derivative of the function/ sinx/(8*cos(x)*sin(x))

Derivative of sinx/(8cos(x)sin(x))

The solution

You have entered [src]

     sin(x)    
---------------
8*cos(x)*sin(x)

$$\frac{\sin{\left(x \right)}}{\sin{\left(x \right)} 8 \cos{\left(x \right)}}$$

sin(x)/(((8*cos(x))*sin(x)))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = 8 \sin{\left(x \right)} \cos{\left(x \right)}$ .

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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
  So, the result is: $- 8 \sin^{2}{\left(x \right)} + 8 \cos^{2}{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{- \left(- 8 \sin^{2}{\left(x \right)} + 8 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} + 8 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{64 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{\sin{\left(x \right)}}{8 \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{\sin{\left(x \right)}}{8 \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{1}{8 \sin{\left(x \right)} \cos{\left(x \right)}} \cos{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)} - 8 \cos^{2}{\left(x \right)}}{64 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                                                                                                                                                                                                    /       2         2                                                   2         2   \                        
           /                                                                      /   1         1   \ /   2         2   \   /   1         1   \ /   2         2   \                             /   2         2   \ / sin(x)    cos(x)\\            |    sin (x) - cos (x)   /   1         1   \ /   2         2   \   sin (x) - cos (x)|                        
           |                                                                      |------- - -------|*\sin (x) - cos (x)/   |------- - -------|*\sin (x) - cos (x)/                           2*\sin (x) - cos (x)/*|------- + -------||          3*|4 + ----------------- + |------- - -------|*\sin (x) - cos (x)/ - -----------------|                        
           |                        /   2         2   \     /   2         2   \   |   2         2   |                       |   2         2   |                         /   2         2   \                         |   3         3   ||            |            2           |   2         2   |                               2        |     /   2         2   \
    1      |     12        12     3*\sin (x) - cos (x)/   3*\sin (x) - cos (x)/   \cos (x)   sin (x)/                       \cos (x)   sin (x)/                       2*\sin (x) - cos (x)/                         \cos (x)   sin (x)/|            \         cos (x)        \cos (x)   sin (x)/                            sin (x)     /   3*\sin (x) - cos (x)/
- ------ + |- ------- + ------- + --------------------- + --------------------- + --------------------------------------- - --------------------------------------- - --------------------- + -----------------------------------------|*sin(x) + --------------------------------------------------------------------------------------- - ---------------------
  sin(x)   |     2         2                4                       4                                2                                         2                            2       2                       cos(x)*sin(x)              |                                                   sin(x)                                                  2             
           \  sin (x)   cos (x)          cos (x)                 sin (x)                          cos (x)                                   sin (x)                      cos (x)*sin (x)                                               /                                                                                                        cos (x)*sin(x)   
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                8

$$\frac{- \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} \cos^{2}{\left(x \right)}} + \frac{3 \left(\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\frac{1}{\cos^{2}{\left(x \right)}} - \frac{1}{\sin^{2}{\left(x \right)}}\right) + \frac{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 4\right)}{\sin{\left(x \right)}} + \left(\frac{2 \left(\frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\frac{1}{\cos^{2}{\left(x \right)}} - \frac{1}{\sin^{2}{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\frac{1}{\cos^{2}{\left(x \right)}} - \frac{1}{\sin^{2}{\left(x \right)}}\right)}{\sin^{2}{\left(x \right)}} + \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{4}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}} + \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{4}{\left(x \right)}} + \frac{12}{\cos^{2}{\left(x \right)}} - \frac{12}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)} - \frac{1}{\sin{\left(x \right)}}}{8}$$

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Derivative of sinx/(8cos(x)sin(x))

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sinx/(8*cos(x)*sin(x))

The graph:

Piecewise:

The solution

Derivative of sinx/(8cos(x)sin(x))