Mister Exam

Other calculators

The derivative of the function/ tg*(ln*(tg(sqrt(3)*x-1)))

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /   /   /  ___      \\\
tan\log\tan\\/ 3 *x - 1///
tan⁡(log⁡(tan⁡(3x−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:

    tan⁡(log⁡(tan⁡(3x−1)))=sin⁡(log⁡(tan⁡(3x−1)))cos⁡(log⁡(tan⁡(3x−1)))\tan{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} = \frac{\sin{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}

  2. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(log⁡(tan⁡(3x−1)))f{\left(x \right)} = \sin{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} and g(x)=cos⁡(log⁡(tan⁡(3x−1)))g{\left(x \right)} = \cos{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=log⁡(tan⁡(3x−1))u = \log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)}.

    2. The derivative of sine is cosine:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddxlog⁡(tan⁡(3x−1))\frac{d}{d x} \log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)}:

      1. Let u=tan⁡(3x−1)u = \tan{\left(\sqrt{3} x - 1 \right)}.

      2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

      3. Then, apply the chain rule. Multiply by ddxtan⁡(3x−1)\frac{d}{d x} \tan{\left(\sqrt{3} x - 1 \right)}:

        1. Let u=3x−1u = \sqrt{3} x - 1.

        2. ddutan⁡(u)=1cos⁡2(u)\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}

        3. Then, apply the chain rule. Multiply by ddx(3x−1)\frac{d}{d x} \left(\sqrt{3} x - 1\right):

          1. Differentiate 3x−1\sqrt{3} x - 1 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: xx goes to 11

              So, the result is: 3\sqrt{3}

            2. The derivative of the constant −1-1 is zero.

            The result is: 3\sqrt{3}

          The result of the chain rule is:

          3cos⁡2(3x−1)\frac{\sqrt{3}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)}}

        The result of the chain rule is:

        3sin⁡2(3x−1)+3cos⁡2(3x−1)cos⁡2(3x−1)tan⁡(3x−1)\frac{\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}}

      The result of the chain rule is:

      (3sin⁡2(3x−1)+3cos⁡2(3x−1))cos⁡(log⁡(tan⁡(3x−1)))cos⁡2(3x−1)tan⁡(3x−1)\frac{\left(\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}\right) \cos{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=log⁡(tan⁡(3x−1))u = \log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)}.

    2. The derivative of cosine is negative sine:

      dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddxlog⁡(tan⁡(3x−1))\frac{d}{d x} \log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)}:

      1. Let u=tan⁡(3x−1)u = \tan{\left(\sqrt{3} x - 1 \right)}.

      2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

      3. Then, apply the chain rule. Multiply by ddxtan⁡(3x−1)\frac{d}{d x} \tan{\left(\sqrt{3} x - 1 \right)}:

        1. Let u=3x−1u = \sqrt{3} x - 1.

        2. ddutan⁡(u)=1cos⁡2(u)\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}

        3. Then, apply the chain rule. Multiply by ddx(3x−1)\frac{d}{d x} \left(\sqrt{3} x - 1\right):

          1. Differentiate 3x−1\sqrt{3} x - 1 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: xx goes to 11

              So, the result is: 3\sqrt{3}

            2. The derivative of the constant −1-1 is zero.

            The result is: 3\sqrt{3}

          The result of the chain rule is:

          3cos⁡2(3x−1)\frac{\sqrt{3}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)}}

        The result of the chain rule is:

        3sin⁡2(3x−1)+3cos⁡2(3x−1)cos⁡2(3x−1)tan⁡(3x−1)\frac{\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}}

      The result of the chain rule is:

      −(3sin⁡2(3x−1)+3cos⁡2(3x−1))sin⁡(log⁡(tan⁡(3x−1)))cos⁡2(3x−1)tan⁡(3x−1)- \frac{\left(\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}\right) \sin{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}}

    Now plug in to the quotient rule:

    (3sin⁡2(3x−1)+3cos⁡2(3x−1))sin⁡2(log⁡(tan⁡(3x−1)))cos⁡2(3x−1)tan⁡(3x−1)+(3sin⁡2(3x−1)+3cos⁡2(3x−1))cos⁡2(log⁡(tan⁡(3x−1)))cos⁡2(3x−1)tan⁡(3x−1)cos⁡2(log⁡(tan⁡(3x−1)))\frac{\frac{\left(\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}\right) \sin^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}} + \frac{\left(\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}\right) \cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}}}{\cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}

  3. Now simplify:

    3cos⁡2(3x−1)cos⁡2(log⁡(tan⁡(3x−1)))tan⁡(3x−1)\frac{\sqrt{3}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} \tan{\left(\sqrt{3} x - 1 \right)}}

The answer is:

3cos⁡2(3x−1)cos⁡2(log⁡(tan⁡(3x−1)))tan⁡(3x−1)\frac{\sqrt{3}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} \tan{\left(\sqrt{3} x - 1 \right)}}

The graph
02468-8-6-4-2-1010-5000050000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  ___ /       2/  ___      \\ /       2/   /   /  ___      \\\\
\/ 3 *\1 + tan \\/ 3 *x - 1//*\1 + tan \log\tan\\/ 3 *x - 1////
---------------------------------------------------------------
                           /  ___      \                       
                        tan\\/ 3 *x - 1/
3(tan⁡2(3x−1)+1)(tan⁡2(log⁡(tan⁡(3x−1)))+1)tan⁡(3x−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)}}
Simplify
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(tan⁡2(3x−1)+1)(tan⁡2(log⁡(tan⁡(3x−1)))+1)(2(tan⁡2(3x−1)+1)tan⁡(log⁡(tan⁡(3x−1)))tan⁡2(3x−1)−tan⁡2(3x−1)+1tan⁡2(3x−1)+2)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)
Simplify
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 /                   /
63(tan⁡2(3x−1)+1)(tan⁡2(log⁡(tan⁡(3x−1)))+1)((tan⁡2(3x−1)+1)2(tan⁡2(log⁡(tan⁡(3x−1)))+1)tan⁡3(3x−1)+2(tan⁡2(3x−1)+1)2tan⁡2(log⁡(tan⁡(3x−1)))tan⁡3(3x−1)−3(tan⁡2(3x−1)+1)2tan⁡(log⁡(tan⁡(3x−1)))tan⁡3(3x−1)+(tan⁡2(3x−1)+1)2tan⁡3(3x−1)+6(tan⁡2(3x−1)+1)tan⁡(log⁡(tan⁡(3x−1)))tan⁡(3x−1)−2(tan⁡2(3x−1)+1)tan⁡(3x−1)+2tan⁡(3x−1))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)
Simplify
    © Mister Exam – Calculator