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Identical expressions
log5(cos(3x))^ four
logarithm of 5( co sinus of e of (3x)) to the power of 4
logarithm of 5( co sinus of e of (3x)) to the power of four
log5(cos(3x))4
log5cos3x4
log5(cos(3x))⁴
log5cos3x^4

The derivative of the function/ log5(cos(3x))^4

Derivative of log5(cos(3x))^4

The solution

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               4
/log(cos(3*x))\ 
|-------------| 
\    log(5)   /

$$\left(\frac{\log{\left(\cos{\left(3 x \right)} \right)}}{\log{\left(5 \right)}}\right)^{4}$$

(log(cos(3*x))/log(5))^4

Detail solution

Let $u = \frac{\log{\left(\cos{\left(3 x \right)} \right)}}{\log{\left(5 \right)}}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(\cos{\left(3 x \right)} \right)}}{\log{\left(5 \right)}}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(3 x \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(3 x \right)}$ :
    1. Let $u = 3 x$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
      The result of the chain rule is:
      
      $- 3 \sin{\left(3 x \right)}$
    The result of the chain rule is:
    
    $- \frac{3 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
  So, the result is: $- \frac{3 \sin{\left(3 x \right)}}{\log{\left(5 \right)} \cos{\left(3 x \right)}}$
The result of the chain rule is:

$- \frac{12 \log{\left(\cos{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{\log{\left(5 \right)}^{4} \cos{\left(3 x \right)}}$
Now simplify:

$- \frac{12 \log{\left(\cos{\left(3 x \right)} \right)}^{3} \tan{\left(3 x \right)}}{\log{\left(5 \right)}^{4}}$

The answer is:

$- \frac{12 \log{\left(\cos{\left(3 x \right)} \right)}^{3} \tan{\left(3 x \right)}}{\log{\left(5 \right)}^{4}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       4                   
    log (cos(3*x))         
-12*--------------*sin(3*x)
          4                
       log (5)             
---------------------------
   cos(3*x)*log(cos(3*x))

$$- \frac{12 \frac{\log{\left(\cos{\left(3 x \right)} \right)}^{4}}{\log{\left(5 \right)}^{4}} \sin{\left(3 x \right)}}{\log{\left(\cos{\left(3 x \right)} \right)} \cos{\left(3 x \right)}}$$

Simplify

The second derivative [src]

                  /                      2           2                   \
      2           |                 3*sin (3*x)   sin (3*x)*log(cos(3*x))|
36*log (cos(3*x))*|-log(cos(3*x)) + ----------- - -----------------------|
                  |                     2                   2            |
                  \                  cos (3*x)           cos (3*x)       /
--------------------------------------------------------------------------
                                    4                                     
                                 log (5)

$$\frac{36 \left(- \frac{\log{\left(\cos{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - \log{\left(\cos{\left(3 x \right)} \right)} + \frac{3 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}\right) \log{\left(\cos{\left(3 x \right)} \right)}^{2}}{\log{\left(5 \right)}^{4}}$$

Simplify

The third derivative [src]

    /                                            2             2              2             2                   \                       
    |       2                               6*sin (3*x)   2*log (cos(3*x))*sin (3*x)   9*sin (3*x)*log(cos(3*x))|                       
108*|- 2*log (cos(3*x)) + 9*log(cos(3*x)) - ----------- - -------------------------- + -------------------------|*log(cos(3*x))*sin(3*x)
    |                                           2                    2                            2             |                       
    \                                        cos (3*x)            cos (3*x)                    cos (3*x)        /                       
----------------------------------------------------------------------------------------------------------------------------------------
                                                                        4                                                               
                                                            cos(3*x)*log (5)

$$\frac{108 \left(- \frac{2 \log{\left(\cos{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - 2 \log{\left(\cos{\left(3 x \right)} \right)}^{2} + \frac{9 \log{\left(\cos{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 9 \log{\left(\cos{\left(3 x \right)} \right)} - \frac{6 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}\right) \log{\left(\cos{\left(3 x \right)} \right)} \sin{\left(3 x \right)}}{\log{\left(5 \right)}^{4} \cos{\left(3 x \right)}}$$

Simplify

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Derivative of log5(cos(3x))^4

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The solution