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1/(sinx+cosx)
The derivative of the function/ 1/(sinx+cosx)

Derivative of 1/(sinx+cosx)

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

The graph:

from to

Piecewise:

The solution

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

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

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

  4. Now simplify:

    −2cos⁡(x+π4)sin⁡(2x)+1- \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(2 x \right)} + 1}

The answer is:

−2cos⁡(x+π4)sin⁡(2x)+1- \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(2 x \right)} + 1}

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 -cos(x) + sin(x) 
------------------
                 2
(sin(x) + cos(x))
sin⁡(x)−cos⁡(x)(sin⁡(x)+cos⁡(x))2\frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}
Simplify
The second derivative [src] 
                        2
    2*(-cos(x) + sin(x)) 
1 + ---------------------
                       2 
      (cos(x) + sin(x))  
-------------------------
     cos(x) + sin(x)
2(sin⁡(x)−cos⁡(x))2(sin⁡(x)+cos⁡(x))2+1sin⁡(x)+cos⁡(x)\frac{\frac{2 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}} + 1}{\sin{\left(x \right)} + \cos{\left(x \right)}}
Simplify
The third derivative [src] 
/                        2\                   
|    6*(-cos(x) + sin(x)) |                   
|5 + ---------------------|*(-cos(x) + sin(x))
|                       2 |                   
\      (cos(x) + sin(x))  /                   
----------------------------------------------
                               2              
              (cos(x) + sin(x))
(6(sin⁡(x)−cos⁡(x))2(sin⁡(x)+cos⁡(x))2+5)(sin⁡(x)−cos⁡(x))(sin⁡(x)+cos⁡(x))2\frac{\left(\frac{6 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}} + 5\right) \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}
Simplify
The graph
Derivative of 1/(sinx+cosx)
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