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

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

Function f() - derivative -N order at the point
v

The graph:

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The solution

You have entered [src]
   /   /   2\\
log\tan\3*x //
$$\log{\left(\tan{\left(3 x^{2} \right)} \right)}$$
log(tan(3*x^2))
Detail solution
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. Let .

      2. The derivative of sine is cosine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      To find :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
    /       2/   2\\
6*x*\1 + tan \3*x //
--------------------
        /   2\      
     tan\3*x /      
$$\frac{6 x \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan{\left(3 x^{2} \right)}}$$
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 /     /
$$6 \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(- \frac{6 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan^{2}{\left(3 x^{2} \right)}} + 12 x^{2} + \frac{1}{\tan{\left(3 x^{2} \right)}}\right)$$
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 /      /
$$108 x \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(\frac{4 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)^{2}}{\tan^{3}{\left(3 x^{2} \right)}} - \frac{8 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan{\left(3 x^{2} \right)}} + 8 x^{2} \tan{\left(3 x^{2} \right)} - \frac{\tan^{2}{\left(3 x^{2} \right)} + 1}{\tan^{2}{\left(3 x^{2} \right)}} + 2\right)$$
The graph
Derivative of y=ln(tg(3x^2))