Derivative of log(sin(x))

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log(sin(x))

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

d              
--(log(sin(x)))
dx

$$\frac{d}{d x} \log{\left(\sin{\left(x \right)} \right)}$$

Transform

Detail solution

Let $u = \sin{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log(sin(x))

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The solution