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

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  • Derivative of:
  • Derivative of x^2+x+1 Derivative of x^2+x+1
  • Derivative of x*2 Derivative of x*2
  • Derivative of (x^2-1)^2 Derivative of (x^2-1)^2
  • Derivative of 1/x^10 Derivative of 1/x^10

  • Identical expressions

  • (- one)/(cos(x)^ two *sin(x)^(two *x))
  • ( minus 1) divide by ( co sinus of e of (x) squared multiply by sinus of (x) to the power of (2 multiply by x))
  • ( minus one) divide by ( co sinus of e of (x) to the power of two multiply by sinus of (x) to the power of (two multiply by x))
  • (-1)/(cos(x)2*sin(x)(2*x))
  • -1/cosx2*sinx2*x
  • (-1)/(cos(x)²*sin(x)^(2*x))
  • (-1)/(cos(x) to the power of 2*sin(x) to the power of (2*x))
  • (-1)/(cos(x)^2sin(x)^(2x))
  • (-1)/(cos(x)2sin(x)(2x))
  • -1/cosx2sinx2x
  • -1/cosx^2sinx^2x
  • (-1) divide by (cos(x)^2*sin(x)^(2*x))

  • Similar expressions

  • (1)/(cos(x)^2*sin(x)^(2*x))
  • (-1)/(cosx^2*sinx^(2*x))
The derivative of the function/ (-1)/(cos(x)^2*sin(x)^(2*x))

Derivative of (-1)/(cos(x)^2*sin(x)^(2*x))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
       -1        
-----------------
   2       2*x   
cos (x)*sin   (x)
$$- \frac{1}{\sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)}}$$ 
d /       -1        \
--|-----------------|
dx|   2       2*x   |
  \cos (x)*sin   (x)/
$$\frac{d}{d x} \left(- \frac{1}{\sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)}}\right)$$
Transform
Detail solution

  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Apply the product rule:

        ; to find :

        1. Let .

        2. Apply the power rule: goes to

        3. Then, apply the chain rule. Multiply by :

          1. The derivative of cosine is negative sine:

          The result of the chain rule is:

        ; to find :

        1. Don't know the steps in finding this derivative.

          But the derivative is

        The result is:

      The result of the chain rule is:

    So, the result is:

  2. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    -4*x    /     2       2*x    /                2*x*cos(x)\        2*x                 \ 
-sin    (x)*|- cos (x)*sin   (x)*|2*log(sin(x)) + ----------| + 2*sin   (x)*cos(x)*sin(x)| 
            \                    \                  sin(x)  /                            / 
-------------------------------------------------------------------------------------------
                                             4                                             
                                          cos (x)
$$- \frac{\left(- \left(\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)}\right) \sin^{- 4 x}{\left(x \right)}}{\cos^{4}{\left(x \right)}}$$
Simplify
The second derivative [src] 
              /                            /                    2   \                             2                                                                                                                                                                                                                                                                       \
              |   2         2         2    |    2*cos(x)   x*cos (x)|     /x*cos(x)              \     2        /x*cos(x)              \                                                                                                                                                                                                                                  |
              |cos (x) - sin (x) + cos (x)*|x - -------- + ---------| - 2*|-------- + log(sin(x))| *cos (x) + 4*|-------- + log(sin(x))|*cos(x)*sin(x)                                                                                                                                                                  /          /x*cos(x)              \       \       |
              |                            |     sin(x)        2    |     \ sin(x)               /              \ sin(x)               /                                                                                                                                                                              2*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*sin(x)|
      -2*x    |                            \                sin (x) /                                                                                      /          /x*cos(x)              \       \ /x*cos(x)              \     /          /x*cos(x)              \       \ /  sin(x)   x*cos(x)              \     \          \ sin(x)               /       /       |
-2*sin    (x)*|--------------------------------------------------------------------------------------------------------------------------------------- + 2*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*|-------- + log(sin(x))| + 2*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*|- ------ + -------- + log(sin(x))| - ----------------------------------------------------|
              \                                                                 cos(x)                                                                     \          \ sin(x)               /       / \ sin(x)               /     \          \ sin(x)               /       / \  cos(x)    sin(x)               /                          cos(x)                       /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                     3                                                                                                                                                                                     
                                                                                                                                                                                  cos (x)
$$- \frac{2 \cdot \left(2 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) + 2 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\right) - \frac{2 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{- 2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right) \sin^{- 2 x}{\left(x \right)}}{\cos^{3}{\left(x \right)}}$$
Simplify
The third derivative [src] 
             /                    /          2             3                \                                                                  3                                                                                             2                                                    /                    2   \     /                    2   \                                                                                                                                                                                                                                                                                                                                                                         /                            /                    2   \                             2                                                   \                                                /                            /                    2   \                             2                                                   \                                                                                                                                           /                            /                    2   \                             2                                                   \                                                                                                                                                                                                                                     \
             |               2    |     3*cos (x)   2*x*cos (x)   2*x*cos(x)|        2    /x*cos(x)              \     /x*cos(x)              \     2                             2    /x*cos(x)              \      /x*cos(x)              \                       2    /x*cos(x)              \ |    2*cos(x)   x*cos (x)|     |    2*cos(x)   x*cos (x)|                                                                                                                                                                                                                                                                                                                                                                         |   2         2         2    |    2*cos(x)   x*cos (x)|     /x*cos(x)              \     2        /x*cos(x)              \              |            /  sin(x)   x*cos(x)              \ |   2         2         2    |    2*cos(x)   x*cos (x)|     /x*cos(x)              \     2        /x*cos(x)              \              |                                                                                                                  /x*cos(x)              \ |   2         2         2    |    2*cos(x)   x*cos (x)|     /x*cos(x)              \     2        /x*cos(x)              \              |                                                                                                                                                                                                                                     |
             |            cos (x)*|-3 - --------- + ----------- + ----------| - 6*cos (x)*|-------- + log(sin(x))| + 4*|-------- + log(sin(x))| *cos (x) + 4*cos(x)*sin(x) + 6*sin (x)*|-------- + log(sin(x))| - 12*|-------- + log(sin(x))| *cos(x)*sin(x) - 6*cos (x)*|-------- + log(sin(x))|*|x - -------- + ---------| + 6*|x - -------- + ---------|*cos(x)*sin(x)                                                                                                                            /                                                                           /x*cos(x)              \       \                                                                                                                 6*|cos (x) - sin (x) + cos (x)*|x - -------- + ---------| - 2*|-------- + log(sin(x))| *cos (x) + 4*|-------- + log(sin(x))|*cos(x)*sin(x)|*sin(x)   2*|- ------ + -------- + log(sin(x))|*|cos (x) - sin (x) + cos (x)*|x - -------- + ---------| - 2*|-------- + log(sin(x))| *cos (x) + 4*|-------- + log(sin(x))|*cos(x)*sin(x)|                                                                                                                6*|-------- + log(sin(x))|*|cos (x) - sin (x) + cos (x)*|x - -------- + ---------| - 2*|-------- + log(sin(x))| *cos (x) + 4*|-------- + log(sin(x))|*cos(x)*sin(x)|         2    /          /x*cos(x)              \       \      /          /x*cos(x)              \       \ /x*cos(x)              \            /          /x*cos(x)              \       \ /  sin(x)   x*cos(x)              \       |
             |                    |         2            3          sin(x)  |             \ sin(x)               /     \ sin(x)               /                                        \ sin(x)               /      \ sin(x)               /                            \ sin(x)               / |     sin(x)        2    |     |     sin(x)        2    |                                                               /                    2   \                                                 |                                  2                   2           2      4*|-------- + log(sin(x))|*sin(x)|                                                                 2                                                 |                            |     sin(x)        2    |     \ sin(x)               /              \ sin(x)               /              |            \  cos(x)    sin(x)               / |                            |     sin(x)        2    |     \ sin(x)               /              \ sin(x)               /              |                                                                                                                  \ sin(x)               / |                            |     sin(x)        2    |     \ sin(x)               /              \ sin(x)               /              |   10*sin (x)*|-sin(x) + |-------- + log(sin(x))|*cos(x)|   16*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*|-------- + log(sin(x))|*sin(x)   4*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*|- ------ + -------- + log(sin(x))|*sin(x)|
     -2*x    |                    \      sin (x)      sin (x)               /                                                                                                                                                                                                                     \                sin (x) /     \                sin (x) /                   /          /x*cos(x)              \       \ |    2*cos(x)   x*cos (x)|     /          /x*cos(x)              \       \ |          /x*cos(x)              \    2*cos(x)   3*sin (x)   x*cos (x)     \ sin(x)               /       |     /x*cos(x)              \            /x*cos(x)              \  /          /x*cos(x)              \       \     \                            \                sin (x) /                                                                                 /                                                \                            \                sin (x) /                                                                                 /     /          /x*cos(x)              \       \ /x*cos(x)              \ /  sin(x)   x*cos(x)              \                              \                            \                sin (x) /                                                                                 /              \          \ sin(x)               /       /      \          \ sin(x)               /       / \ sin(x)               /            \          \ sin(x)               /       / \  cos(x)    sin(x)               /       |
2*sin    (x)*|-2*sin(x) + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + 2*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*|x - -------- + ---------| + 2*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*|1 + x + 2*|-------- + log(sin(x))|  - -------- + --------- + --------- - ---------------------------------| + 2*|-------- + log(sin(x))|*cos(x) + 8*|-------- + log(sin(x))| *|-sin(x) + |-------- + log(sin(x))|*cos(x)| - -------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + 4*|-sin(x) + |-------- + log(sin(x))|*cos(x)|*|-------- + log(sin(x))|*|- ------ + -------- + log(sin(x))| + -------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------ - ------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------|
             |                                                                                                                                                                                 cos(x)                                                                                                                                                                         \          \ sin(x)               /       / |     sin(x)        2    |     \          \ sin(x)               /       / |          \ sin(x)               /     sin(x)        2           2                     cos(x)             |     \ sin(x)               /            \ sin(x)               /  \          \ sin(x)               /       /                                                                           2                                                                                                                                                                 cos(x)                                                                                         \          \ sin(x)               /       / \ sin(x)               / \  cos(x)    sin(x)               /                                                                                  cos(x)                                                                                                            2                                                                  cos(x)                                                                                cos(x)                                         |
             \                                                                                                                                                                                                                                                                                                                                                                                                            \                sin (x) /                                                 \                                                  cos (x)     sin (x)                                     /                                                                                                                                                                                      cos (x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      cos (x)                                                                                                                                                                                                    /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            cos (x)
$$\frac{2 \cdot \left(8 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} + 4 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\right) - \frac{16 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) - \frac{4 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} - \frac{4 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) + \frac{10 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{6 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \left(- 2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos{\left(x \right)}} + 2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} + \frac{2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\right) \left(- 2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos{\left(x \right)}} - \frac{6 \left(- 2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{4 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{3} \cos^{2}{\left(x \right)} - 12 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 6 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos^{2}{\left(x \right)} + 6 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{2}{\left(x \right)} - 6 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \cos^{2}{\left(x \right)} + 6 \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)} + 4 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)}} - 2 \sin{\left(x \right)}\right) \sin^{- 2 x}{\left(x \right)}}{\cos^{3}{\left(x \right)}}$$
Simplify
The graph
Derivative of (-1)/(cos(x)^2*sin(x)^(2*x))
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