Derivative of log(sin(x)+cos(x))

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log(sin(x) + cos(x))

$$\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}$$

log(sin(x) + cos(x))

Detail solution

Let $u = \sin{\left(x \right)} + \cos{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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:

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

$\frac{1}{\tan{\left(x + \frac{\pi}{4} \right)}}$

The answer is:

$\frac{1}{\tan{\left(x + \frac{\pi}{4} \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

-sin(x) + cos(x)
----------------
sin(x) + cos(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log(sin(x)+cos(x))

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