Derivative of -ln(-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

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = - \sin{\left(x \right)} - \cos{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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 a constant times a function is the constant times the derivative of the function.
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      So, the result is: $- \cos{\left(x \right)}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      So, the result is: $\sin{\left(x \right)}$
    The result is: $\sin{\left(x \right)} - \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{- \sin{\left(x \right)} - \cos{\left(x \right)}}$
So, the result 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

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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