Derivative of log3(sinx)

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log(sin(x))
-----------
   log(3)

$$\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}}$$

log(sin(x))/log(3)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. 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:
  
  $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
So, the result is: $\frac{\cos{\left(x \right)}}{\log{\left(3 \right)} \sin{\left(x \right)}}$
Now simplify:

$\frac{1}{\log{\left(3 \right)} \tan{\left(x \right)}}$

The answer is:

$\frac{1}{\log{\left(3 \right)} \tan{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    cos(x)   
-------------
log(3)*sin(x)

$$\frac{\cos{\left(x \right)}}{\log{\left(3 \right)} \sin{\left(x \right)}}$$

Simplify

The second derivative [src]

 /       2   \ 
 |    cos (x)| 
-|1 + -------| 
 |       2   | 
 \    sin (x)/ 
---------------
     log(3)

$$- \frac{1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}}{\log{\left(3 \right)}}$$

Simplify

The third derivative [src]

  /       2   \       
  |    cos (x)|       
2*|1 + -------|*cos(x)
  |       2   |       
  \    sin (x)/       
----------------------
    log(3)*sin(x)

$$\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\log{\left(3 \right)} \sin{\left(x \right)}}$$

Simplify

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Derivative of log3(sinx)

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The solution