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ln(cos(3x))*tan^ two (3x)
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The derivative of the function/ ln(cos(3x))*tan^2(3x)

Derivative of ln(cos(3x))*tan^2(3x)

The solution

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                 2     
log(cos(3*x))*tan (3*x)

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

log(cos(3*x))*tan(3*x)^2

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \log{\left(\cos{\left(3 x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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.
      1. 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:
  
  $- \frac{3 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
$g{\left(x \right)} = \tan^{2}{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \tan{\left(3 x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos{\left(3 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
      The result of the chain rule is:
      
      $3 \cos{\left(3 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    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:
      
      $- 3 \sin{\left(3 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
  The result of the chain rule is:
  
  $\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \tan{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
The result is: $\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \log{\left(\cos{\left(3 x \right)} \right)} \tan{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - \frac{3 \sin{\left(3 x \right)} \tan^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
Now simplify:

$\frac{6 \log{\left(\cos{\left(3 x \right)} \right)} \sin{\left(3 x \right)}}{\cos^{3}{\left(3 x \right)}} - 3 \tan^{3}{\left(3 x \right)}$

The answer is:

$\frac{6 \log{\left(\cos{\left(3 x \right)} \right)} \sin{\left(3 x \right)}}{\cos^{3}{\left(3 x \right)}} - 3 \tan^{3}{\left(3 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                                2              
/         2     \                          3*tan (3*x)*sin(3*x)
\6 + 6*tan (3*x)/*log(cos(3*x))*tan(3*x) - --------------------
                                                 cos(3*x)

$$\left(6 \tan^{2}{\left(3 x \right)} + 6\right) \log{\left(\cos{\left(3 x \right)} \right)} \tan{\left(3 x \right)} - \frac{3 \sin{\left(3 x \right)} \tan^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)}}$$

Simplify

The second derivative [src]

  /            /       2     \                                                         /       2     \                  \
  |     2      |    sin (3*x)|     /       2     \ /         2     \                 4*\1 + tan (3*x)/*sin(3*x)*tan(3*x)|
9*|- tan (3*x)*|1 + ---------| + 2*\1 + tan (3*x)/*\1 + 3*tan (3*x)/*log(cos(3*x)) - -----------------------------------|
  |            |       2     |                                                                     cos(3*x)             |
  \            \    cos (3*x)/                                                                                          /

$$9 \left(- \left(\frac{\sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 1\right) \tan^{2}{\left(3 x \right)} + 2 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 1\right) \log{\left(\cos{\left(3 x \right)} \right)} - \frac{4 \left(\tan^{2}{\left(3 x \right)} + 1\right) \sin{\left(3 x \right)} \tan{\left(3 x \right)}}{\cos{\left(3 x \right)}}\right)$$

Simplify

The third derivative [src]

   /                                                         /       2     \                                                                                                                     \
   |                                                  2      |    sin (3*x)|                                                                                                                     |
   |                                               tan (3*x)*|1 + ---------|*sin(3*x)                                                                                                            |
   |                    /       2     \                      |       2     |              /       2     \ /         2     \                                                                      |
   |    /       2     \ |    sin (3*x)|                      \    cos (3*x)/            3*\1 + tan (3*x)/*\1 + 3*tan (3*x)/*sin(3*x)     /       2     \ /         2     \                       |
54*|- 3*\1 + tan (3*x)/*|1 + ---------|*tan(3*x) - ---------------------------------- - -------------------------------------------- + 4*\1 + tan (3*x)/*\2 + 3*tan (3*x)/*log(cos(3*x))*tan(3*x)|
   |                    |       2     |                         cos(3*x)                                  cos(3*x)                                                                               |
   \                    \    cos (3*x)/                                                                                                                                                          /

$$54 \left(- 3 \left(\frac{\sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 1\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{\left(\frac{\sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 1\right) \sin{\left(3 x \right)} \tan^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)}} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 1\right) \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}} + 4 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 2\right) \log{\left(\cos{\left(3 x \right)} \right)} \tan{\left(3 x \right)}\right)$$

Simplify

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Identical expressions

Derivative of ln(cos(3x))*tan^2(3x)

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The solution