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

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   /   /   2\\
log\tan\3*x //
log⁡(tan⁡(3x2))\log{\left(\tan{\left(3 x^{2} \right)} \right)} 
log(tan(3*x^2))
Detail solution

  1. Let u=tan⁡(3x2)u = \tan{\left(3 x^{2} \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⁡(3x2)\frac{d}{d x} \tan{\left(3 x^{2} \right)}:

    1. Rewrite the function to be differentiated:

      tan⁡(3x2)=sin⁡(3x2)cos⁡(3x2)\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:

      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⁡(3x2)f{\left(x \right)} = \sin{\left(3 x^{2} \right)} and g(x)=cos⁡(3x2)g{\left(x \right)} = \cos{\left(3 x^{2} \right)}.

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

      1. Let u=3x2u = 3 x^{2}.

      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 ddx3x2\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: x2x^{2} goes to 2x2 x

          So, the result is: 6x6 x

        The result of the chain rule is:

        6xcos⁡(3x2)6 x \cos{\left(3 x^{2} \right)}

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

      1. Let u=3x2u = 3 x^{2}.

      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 ddx3x2\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: x2x^{2} goes to 2x2 x

          So, the result is: 6x6 x

        The result of the chain rule is:

        −6xsin⁡(3x2)- 6 x \sin{\left(3 x^{2} \right)}

      Now plug in to the quotient rule:

      6xsin⁡2(3x2)+6xcos⁡2(3x2)cos⁡2(3x2)\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:

    6xsin⁡2(3x2)+6xcos⁡2(3x2)cos⁡2(3x2)tan⁡(3x2)\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)}}

  4. Now simplify:

    12xsin⁡(6x2)\frac{12 x}{\sin{\left(6 x^{2} \right)}}

The answer is:

12xsin⁡(6x2)\frac{12 x}{\sin{\left(6 x^{2} \right)}}

The graph
02468-8-6-4-2-1010-50005000
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 /
6x(tan⁡2(3x2)+1)tan⁡(3x2)\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(tan⁡2(3x2)+1)(−6x2(tan⁡2(3x2)+1)tan⁡2(3x2)+12x2+1tan⁡(3x2))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 /      /
108x(tan⁡2(3x2)+1)(4x2(tan⁡2(3x2)+1)2tan⁡3(3x2)−8x2(tan⁡2(3x2)+1)tan⁡(3x2)+8x2tan⁡(3x2)−tan⁡2(3x2)+1tan⁡2(3x2)+2)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
The graph
Derivative of y=ln(tg(3x^2))
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