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Derivative of tg*(ln*(tg(sqrt(3)*x-1)))

Function f() - derivative -N order at the point
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Piecewise:

The solution

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

      2. The derivative of is .

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

        1. Let .

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

          1. Differentiate term by term:

            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:

            2. The derivative of the constant is zero.

            The result is:

          The result of the chain rule is:

        The result of the chain rule 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. Let .

      2. The derivative of is .

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

        1. Let .

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

          1. Differentiate term by term:

            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:

            2. The derivative of the constant is zero.

            The result is:

          The result of the chain rule is:

        The result of the chain rule is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  3. Now simplify:


The answer is:

The graph
The first derivative [src]
  ___ /       2/  ___      \\ /       2/   /   /  ___      \\\\
\/ 3 *\1 + tan \\/ 3 *x - 1//*\1 + tan \log\tan\\/ 3 *x - 1////
---------------------------------------------------------------
                           /  ___      \                       
                        tan\\/ 3 *x - 1/                       
$$\frac{\sqrt{3} \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right) \left(\tan^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} + 1\right)}{\tan{\left(\sqrt{3} x - 1 \right)}}$$
The second derivative [src]
                                                              /           2/         ___\     /       2/         ___\\    /   /   /         ___\\\\
  /       2/         ___\\ /       2/   /   /         ___\\\\ |    1 + tan \-1 + x*\/ 3 /   2*\1 + tan \-1 + x*\/ 3 //*tan\log\tan\-1 + x*\/ 3 ///|
3*\1 + tan \-1 + x*\/ 3 //*\1 + tan \log\tan\-1 + x*\/ 3 ////*|2 - ---------------------- + ------------------------------------------------------|
                                                              |         2/         ___\                          2/         ___\                  |
                                                              \      tan \-1 + x*\/ 3 /                       tan \-1 + x*\/ 3 /                  /
$$3 \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right) \left(\tan^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right) \tan{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\tan^{2}{\left(\sqrt{3} x - 1 \right)}} - \frac{\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1}{\tan^{2}{\left(\sqrt{3} x - 1 \right)}} + 2\right)$$
The third derivative [src]
                                                                    /                                              2                                                        2                                                                2                                                         2                                                                                      \
                                                                    |                      /       2/         ___\\      /       2/         ___\\   /       2/         ___\\  /       2/   /   /         ___\\\\     /       2/         ___\\     /   /   /         ___\\\     /       2/         ___\\     2/   /   /         ___\\\     /       2/         ___\\    /   /   /         ___\\\|
    ___ /       2/         ___\\ /       2/   /   /         ___\\\\ |     /         ___\   \1 + tan \-1 + x*\/ 3 //    2*\1 + tan \-1 + x*\/ 3 //   \1 + tan \-1 + x*\/ 3 // *\1 + tan \log\tan\-1 + x*\/ 3 ////   3*\1 + tan \-1 + x*\/ 3 // *tan\log\tan\-1 + x*\/ 3 ///   2*\1 + tan \-1 + x*\/ 3 // *tan \log\tan\-1 + x*\/ 3 ///   6*\1 + tan \-1 + x*\/ 3 //*tan\log\tan\-1 + x*\/ 3 ///|
6*\/ 3 *\1 + tan \-1 + x*\/ 3 //*\1 + tan \log\tan\-1 + x*\/ 3 ////*|2*tan\-1 + x*\/ 3 / + ------------------------- - -------------------------- + ------------------------------------------------------------ - ------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------|
                                                                    |                             3/         ___\             /         ___\                                3/         ___\                                              3/         ___\                                           3/         ___\                                           /         ___\                   |
                                                                    \                          tan \-1 + x*\/ 3 /          tan\-1 + x*\/ 3 /                             tan \-1 + x*\/ 3 /                                           tan \-1 + x*\/ 3 /                                        tan \-1 + x*\/ 3 /                                        tan\-1 + x*\/ 3 /                   /
$$6 \sqrt{3} \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right) \left(\tan^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right)^{2} \left(\tan^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} + 1\right)}{\tan^{3}{\left(\sqrt{3} x - 1 \right)}} + \frac{2 \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right)^{2} \tan^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\tan^{3}{\left(\sqrt{3} x - 1 \right)}} - \frac{3 \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right)^{2} \tan{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\tan^{3}{\left(\sqrt{3} x - 1 \right)}} + \frac{\left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right)^{2}}{\tan^{3}{\left(\sqrt{3} x - 1 \right)}} + \frac{6 \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right) \tan{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\tan{\left(\sqrt{3} x - 1 \right)}} - \frac{2 \left(\tan^{2}{\left(\sqrt{3} x - 1 \right)} + 1\right)}{\tan{\left(\sqrt{3} x - 1 \right)}} + 2 \tan{\left(\sqrt{3} x - 1 \right)}\right)$$