Derivative of ln(tan(3x))

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log(tan(3*x))

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

d                
--(log(tan(3*x)))
dx

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

Transform

Detail solution

Let $u = \tan{\left(3 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result of the chain rule is:
    
    $3 \cos{\left(3 x \right)}$
  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.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result of the chain rule is:
    
    $- 3 \sin{\left(3 x \right)}$
  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:

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

$\frac{6}{\sin{\left(6 x \right)}}$

The answer is:

$\frac{6}{\sin{\left(6 x \right)}}$

The graph

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The first derivative [src]

         2     
3 + 3*tan (3*x)
---------------
    tan(3*x)

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

Simplify

The second derivative [src]

  /                                 2\
  |                  /       2     \ |
  |         2        \1 + tan (3*x)/ |
9*|2 + 2*tan (3*x) - ----------------|
  |                        2         |
  \                     tan (3*x)    /

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

Simplify

The third derivative [src]

                   /                            2                    \
                   |             /       2     \      /       2     \|
   /       2     \ |             \1 + tan (3*x)/    2*\1 + tan (3*x)/|
54*\1 + tan (3*x)/*|2*tan(3*x) + ---------------- - -----------------|
                   |                   3                 tan(3*x)    |
                   \                tan (3*x)                        /

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

Simplify

The graph

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Identical expressions

Derivative of ln(tan(3x))

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The solution