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The derivative of the function/ 1/(cos(x)sin(x))

Derivative of 1/(cos(x)sin(x))

The solution

You have entered [src]

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

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

1/(cos(x)*sin(x))

Detail solution

Let $u = \sin{\left(x \right)} \cos{\left(x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)} \cos{\left(x \right)}$ :
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: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1       /   2         2   \
-------------*\sin (x) - cos (x)/
cos(x)*sin(x)                    
---------------------------------
          cos(x)*sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 1/(cos(x)sin(x))

The graph:

Piecewise:

The solution