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The derivative of the function/ 1/((sin(x)+cos(x))^2)

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

The solution

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        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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -(-2*sin(x) + 2*cos(x))       
------------------------------------
                                   2
(sin(x) + cos(x))*(sin(x) + cos(x))

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

Simplify

The second derivative [src]

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

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

Simplify

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))

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

Simplify

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

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The solution