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Derivative of 1/((sin(x)+cos(x))^2)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
        1         
------------------
                 2
(sin(x) + cos(x)) 
$$\frac{1}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}$$
1/((sin(x) + cos(x))^2)
Detail solution
  1. Let .

  2. Apply the power rule: goes to

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

    1. Let .

    2. Apply the power rule: goes to

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

      1. Differentiate term by term:

        1. The derivative of sine is cosine:

        2. The derivative of cosine is negative sine:

        The result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
      -(-2*sin(x) + 2*cos(x))       
------------------------------------
                                   2
(sin(x) + cos(x))*(sin(x) + cos(x)) 
$$- \frac{- 2 \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}$$
The second derivative [src]
  /                        2\
  |    3*(-cos(x) + sin(x)) |
2*|1 + ---------------------|
  |                       2 |
  \      (cos(x) + sin(x))  /
-----------------------------
                       2     
      (cos(x) + sin(x))      
$$\frac{2 \left(\frac{3 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}} + 1\right)}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}$$
The third derivative [src]
  /                        2\                   
  |    3*(-cos(x) + sin(x)) |                   
8*|2 + ---------------------|*(-cos(x) + sin(x))
  |                       2 |                   
  \      (cos(x) + sin(x))  /                   
------------------------------------------------
                                3               
               (cos(x) + sin(x))                
$$\frac{8 \left(\frac{3 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}} + 2\right) \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{3}}$$