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Identical expressions
y=ln(tg(3x^ two))
y equally ln(tg(3x squared ))
y equally ln(tg(3x to the power of two))
y=ln(tg(3x2))
y=lntg3x2
y=ln(tg(3x²))
y=ln(tg(3x to the power of 2))
y=lntg3x^2

The derivative of the function/ y=ln(tg(3x^2))

Derivative of y=ln(tg(3x^2))

The solution

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   /   /   2\\
log\tan\3*x //

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

log(tan(3*x^2))

Detail solution

Let $u = \tan{\left(3 x^{2} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 x^{2} \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(3 x^{2} \right)} = \frac{\sin{\left(3 x^{2} \right)}}{\cos{\left(3 x^{2} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(3 x^{2} \right)}$ and $g{\left(x \right)} = \cos{\left(3 x^{2} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 3 x^{2}$ .
  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 $\frac{d}{d x} 3 x^{2}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $6 x$
    The result of the chain rule is:
    
    $6 x \cos{\left(3 x^{2} \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 3 x^{2}$ .
  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 $\frac{d}{d x} 3 x^{2}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $6 x$
    The result of the chain rule is:
    
    $- 6 x \sin{\left(3 x^{2} \right)}$
  Now plug in to the quotient rule:
  
  $\frac{6 x \sin^{2}{\left(3 x^{2} \right)} + 6 x \cos^{2}{\left(3 x^{2} \right)}}{\cos^{2}{\left(3 x^{2} \right)}}$
The result of the chain rule is:

$\frac{6 x \sin^{2}{\left(3 x^{2} \right)} + 6 x \cos^{2}{\left(3 x^{2} \right)}}{\cos^{2}{\left(3 x^{2} \right)} \tan{\left(3 x^{2} \right)}}$
Now simplify:

$\frac{12 x}{\sin{\left(6 x^{2} \right)}}$

The answer is:

$\frac{12 x}{\sin{\left(6 x^{2} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /       2/   2\\
6*x*\1 + tan \3*x //
--------------------
        /   2\      
     tan\3*x /

$$\frac{6 x \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan{\left(3 x^{2} \right)}}$$

Simplify

The second derivative [src]

                   /                       2 /       2/   2\\\
  /       2/   2\\ |    1           2   6*x *\1 + tan \3*x //|
6*\1 + tan \3*x //*|--------- + 12*x  - ---------------------|
                   |   /   2\                    2/   2\     |
                   \tan\3*x /                 tan \3*x /     /

$$6 \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(- \frac{6 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan^{2}{\left(3 x^{2} \right)}} + 12 x^{2} + \frac{1}{\tan{\left(3 x^{2} \right)}}\right)$$

Simplify

The third derivative [src]

                       /                                                                                   2\
                       |           2/   2\                       2 /       2/   2\\      2 /       2/   2\\ |
      /       2/   2\\ |    1 + tan \3*x /      2    /   2\   8*x *\1 + tan \3*x //   4*x *\1 + tan \3*x // |
108*x*\1 + tan \3*x //*|2 - -------------- + 8*x *tan\3*x / - --------------------- + ----------------------|
                       |         2/   2\                               /   2\                  3/   2\      |
                       \      tan \3*x /                            tan\3*x /               tan \3*x /      /

$$108 x \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(\frac{4 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)^{2}}{\tan^{3}{\left(3 x^{2} \right)}} - \frac{8 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan{\left(3 x^{2} \right)}} + 8 x^{2} \tan{\left(3 x^{2} \right)} - \frac{\tan^{2}{\left(3 x^{2} \right)} + 1}{\tan^{2}{\left(3 x^{2} \right)}} + 2\right)$$

Simplify

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Derivative of y=ln(tg(3x^2))

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The solution