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log(sin(x))
The derivative of the function/ log(sin(x))

Derivative of log(sin(x))

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

The graph:

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The solution

You have entered [src] 
log(sin(x))
log⁡(sin⁡(x))\log{\left(\sin{\left(x \right)} \right)} 
d              
--(log(sin(x)))
dx
ddxlog⁡(sin⁡(x))\frac{d}{d x} \log{\left(\sin{\left(x \right)} \right)}
Transform
Detail solution

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

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result of the chain rule is:

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

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-200200
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
cos(x)
------
sin(x)
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+cos⁡2(x)sin⁡2(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)
2⋅(1+cos⁡2(x)sin⁡2(x))cos⁡(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
The graph
Derivative of log(sin(x))
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