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

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

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

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The solution

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

  1. Let u=(sin⁡(x)+cos⁡(x))2u = \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}.

  2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

  3. Then, apply the chain rule. Multiply by ddx(sin⁡(x)+cos⁡(x))2\frac{d}{d x} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}:

    1. Let u=sin⁡(x)+cos⁡(x)u = \sin{\left(x \right)} + \cos{\left(x \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddx(sin⁡(x)+cos⁡(x))\frac{d}{d x} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right):

      1. Differentiate sin⁡(x)+cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)} term by term:

        1. The derivative of sine is cosine:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        2. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        The result is: −sin⁡(x)+cos⁡(x)- \sin{\left(x \right)} + \cos{\left(x \right)}

      The result of the chain rule is:

      (−sin⁡(x)+cos⁡(x))(2sin⁡(x)+2cos⁡(x))\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:

    −(−sin⁡(x)+cos⁡(x))(2sin⁡(x)+2cos⁡(x))(sin⁡(x)+cos⁡(x))4- \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}}

  4. Now simplify:

    −cos⁡(x+π4)sin⁡3(x+π4)- \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{\sin^{3}{\left(x + \frac{\pi}{4} \right)}}

The answer is:

−cos⁡(x+π4)sin⁡3(x+π4)- \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{\sin^{3}{\left(x + \frac{\pi}{4} \right)}}

The graph
02468-8-6-4-2-1010-5000001000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      -(-2*sin(x) + 2*cos(x))       
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                                   2
(sin(x) + cos(x))*(sin(x) + cos(x))
−−2sin⁡(x)+2cos⁡(x)(sin⁡(x)+cos⁡(x))(sin⁡(x)+cos⁡(x))2- \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))  /
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                       2     
      (cos(x) + sin(x))
2(3(sin⁡(x)−cos⁡(x))2(sin⁡(x)+cos⁡(x))2+1)(sin⁡(x)+cos⁡(x))2\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))
8(3(sin⁡(x)−cos⁡(x))2(sin⁡(x)+cos⁡(x))2+2)(sin⁡(x)−cos⁡(x))(sin⁡(x)+cos⁡(x))3\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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