Derivative of ln(sinx-cosx)

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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 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:

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

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

The answer is:

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

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)) /

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

Simplify

3-я производная [src]

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

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

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)

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

Simplify

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Derivative of ln(sinx-cosx)

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The solution