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Identical expressions
y=(log3(sinx)^ two)
y equally ( logarithm of 3( sinus of x) squared )
y equally ( logarithm of 3( sinus of x) to the power of two)
y=(log3(sinx)2)
y=log3sinx2
y=(log3(sinx)²)
y=(log3(sinx) to the power of 2)
y=log3sinx^2

The derivative of the function/ y=(log3(sinx)^2)

Derivative of y=(log3(sinx)^2)

The solution

You have entered [src]

             2
/log(sin(x))\ 
|-----------| 
\   log(3)  /

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

  /             2\
d |/log(sin(x))\ |
--||-----------| |
dx\\   log(3)  / /

$$\frac{d}{d x} \left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}}\right)^{2}$$

Transform

Detail solution

Let $u = \frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}}$ :
1. 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: $\frac{\cos{\left(x \right)}}{\log{\left(3 \right)} \sin{\left(x \right)}}$
The result of the chain rule is:

$\frac{2 \log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\log{\left(3 \right)}^{2} \sin{\left(x \right)}}$
Now simplify:

$\frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}^{2} \tan{\left(x \right)}}$

The answer is:

$\frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}^{2} \tan{\left(x \right)}}$

The first derivative [src]

     2               
  log (sin(x))       
2*------------*cos(x)
       2             
    log (3)          
---------------------
  log(sin(x))*sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=(log3(sinx)^2)

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The solution