Derivative of y=sqrt(ln(sin(3x)))

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  _______________
\/ log(sin(3*x))

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

d /  _______________\
--\\/ log(sin(3*x)) /
dx

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

Transform

Detail solution

Let $u = \log{\left(\sin{\left(3 x \right)} \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\sin{\left(3 x \right)} \right)}$ :
1. Let $u = \sin{\left(3 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
  1. Let $u = 3 x$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result of the chain rule is:
    
    $3 \cos{\left(3 x \right)}$
  The result of the chain rule is:
  
  $\frac{3 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

   /         2                  2            \
   |    2*cos (3*x)          cos (3*x)       |
-9*|2 + ----------- + -----------------------|
   |        2                          2     |
   \     sin (3*x)    log(sin(3*x))*sin (3*x)/
----------------------------------------------
                 _______________              
             4*\/ log(sin(3*x))

$$- \frac{9 \cdot \left(2 + \frac{2 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{\cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)}}\right)}{4 \sqrt{\log{\left(\sin{\left(3 x \right)} \right)}}}$$

Simplify

The third derivative [src]

   /                         2                    2                           2             \         
   |           3          cos (3*x)          3*cos (3*x)                 3*cos (3*x)        |         
27*|1 + --------------- + --------- + ------------------------- + --------------------------|*cos(3*x)
   |    4*log(sin(3*x))      2                           2             2              2     |         
   \                      sin (3*x)   4*log(sin(3*x))*sin (3*x)   8*log (sin(3*x))*sin (3*x)/         
------------------------------------------------------------------------------------------------------
                                        _______________                                               
                                      \/ log(sin(3*x)) *sin(3*x)

$$\frac{27 \cdot \left(1 + \frac{\cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{3}{4 \log{\left(\sin{\left(3 x \right)} \right)}} + \frac{3 \cos^{2}{\left(3 x \right)}}{4 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)}} + \frac{3 \cos^{2}{\left(3 x \right)}}{8 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)}}\right) \cos{\left(3 x \right)}}{\sqrt{\log{\left(\sin{\left(3 x \right)} \right)}} \sin{\left(3 x \right)}}$$

Simplify

The graph

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Derivative of y=sqrt(ln(sin(3x)))

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