Derivative of y=ln³(sin7x)

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   3          
log (sin(7*x))

$$\log{\left(\sin{\left(7 x \right)} \right)}^{3}$$

d /   3          \
--\log (sin(7*x))/
dx

$$\frac{d}{d x} \log{\left(\sin{\left(7 x \right)} \right)}^{3}$$

Transform

Detail solution

Let $u = \log{\left(\sin{\left(7 x \right)} \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\sin{\left(7 x \right)} \right)}$ :
1. Let $u = \sin{\left(7 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(7 x \right)}$ :
  1. Let $u = 7 x$ .
  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 $\frac{d}{d x} 7 x$ :
    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: $7$
    The result of the chain rule is:
    
    $7 \cos{\left(7 x \right)}$
  The result of the chain rule is:
  
  $\frac{7 \cos{\left(7 x \right)}}{\sin{\left(7 x \right)}}$
The result of the chain rule is:

$\frac{21 \log{\left(\sin{\left(7 x \right)} \right)}^{2} \cos{\left(7 x \right)}}{\sin{\left(7 x \right)}}$
Now simplify:

$\frac{21 \log{\left(\sin{\left(7 x \right)} \right)}^{2}}{\tan{\left(7 x \right)}}$

The answer is:

$\frac{21 \log{\left(\sin{\left(7 x \right)} \right)}^{2}}{\tan{\left(7 x \right)}}$

The graph

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The first derivative [src]

      2                   
21*log (sin(7*x))*cos(7*x)
--------------------------
         sin(7*x)

$$\frac{21 \log{\left(\sin{\left(7 x \right)} \right)}^{2} \cos{\left(7 x \right)}}{\sin{\left(7 x \right)}}$$

Simplify

The second derivative [src]

    /                      2           2                   \              
    |                 2*cos (7*x)   cos (7*x)*log(sin(7*x))|              
147*|-log(sin(7*x)) + ----------- - -----------------------|*log(sin(7*x))
    |                     2                   2            |              
    \                  sin (7*x)           sin (7*x)       /

$$147 \left(- \log{\left(\sin{\left(7 x \right)} \right)} - \frac{\log{\left(\sin{\left(7 x \right)} \right)} \cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}} + \frac{2 \cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}}\right) \log{\left(\sin{\left(7 x \right)} \right)}$$

Simplify

The third derivative [src]

     /                                      2           2         2                  2                   \         
     |   2                               cos (7*x)   cos (7*x)*log (sin(7*x))   3*cos (7*x)*log(sin(7*x))|         
2058*|log (sin(7*x)) - 3*log(sin(7*x)) + --------- + ------------------------ - -------------------------|*cos(7*x)
     |                                      2                  2                           2             |         
     \                                   sin (7*x)          sin (7*x)                   sin (7*x)        /         
-------------------------------------------------------------------------------------------------------------------
                                                      sin(7*x)

$$\frac{2058 \left(\log{\left(\sin{\left(7 x \right)} \right)}^{2} + \frac{\log{\left(\sin{\left(7 x \right)} \right)}^{2} \cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}} - 3 \log{\left(\sin{\left(7 x \right)} \right)} - \frac{3 \log{\left(\sin{\left(7 x \right)} \right)} \cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}} + \frac{\cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}}\right) \cos{\left(7 x \right)}}{\sin{\left(7 x \right)}}$$

Simplify

3-я производная [src]

     /                                      2           2         2                  2                   \         
     |   2                               cos (7*x)   cos (7*x)*log (sin(7*x))   3*cos (7*x)*log(sin(7*x))|         
2058*|log (sin(7*x)) - 3*log(sin(7*x)) + --------- + ------------------------ - -------------------------|*cos(7*x)
     |                                      2                  2                           2             |         
     \                                   sin (7*x)          sin (7*x)                   sin (7*x)        /         
-------------------------------------------------------------------------------------------------------------------
                                                      sin(7*x)

$$\frac{2058 \left(\log{\left(\sin{\left(7 x \right)} \right)}^{2} + \frac{\log{\left(\sin{\left(7 x \right)} \right)}^{2} \cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}} - 3 \log{\left(\sin{\left(7 x \right)} \right)} - \frac{3 \log{\left(\sin{\left(7 x \right)} \right)} \cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}} + \frac{\cos^{2}{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}}\right) \cos{\left(7 x \right)}}{\sin{\left(7 x \right)}}$$

Simplify

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Derivative of y=ln³(sin7x)

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