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y=ln(tgx^2)

Derivative of y=ln(tgx^2)

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

You have entered [src]
   /   2   \
log\tan (x)/
log(tan2(x))\log{\left(\tan^{2}{\left(x \right)} \right)}
log(tan(x)^2)
Detail solution
  1. Let u=tan2(x)u = \tan^{2}{\left(x \right)}.

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

  3. Then, apply the chain rule. Multiply by ddxtan2(x)\frac{d}{d x} \tan^{2}{\left(x \right)}:

    1. Let u=tan(x)u = \tan{\left(x \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

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

      1. Rewrite the function to be differentiated:

        tan(x)=sin(x)cos(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \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(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos(x)g{\left(x \right)} = \cos{\left(x \right)}.

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

        1. The derivative of sine is cosine:

          ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

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

        1. The derivative of cosine is negative sine:

          ddxcos(x)=sin(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Now plug in to the quotient rule:

        sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      The result of the chain rule is:

      2(sin2(x)+cos2(x))tan(x)cos2(x)\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result of the chain rule is:

    2(sin2(x)+cos2(x))cos2(x)tan(x)\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}

  4. Now simplify:

    4sin(2x)\frac{4}{\sin{\left(2 x \right)}}


The answer is:

4sin(2x)\frac{4}{\sin{\left(2 x \right)}}

The graph
02468-8-6-4-2-1010-500500
The first derivative [src]
         2   
2 + 2*tan (x)
-------------
    tan(x)   
2tan2(x)+2tan(x)\frac{2 \tan^{2}{\left(x \right)} + 2}{\tan{\left(x \right)}}
The second derivative [src]
  /                             2\
  |                /       2   \ |
  |         2      \1 + tan (x)/ |
2*|2 + 2*tan (x) - --------------|
  |                      2       |
  \                   tan (x)    /
2((tan2(x)+1)2tan2(x)+2tan2(x)+2)2 \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right)
The third derivative [src]
                /                        2                  \
                |           /       2   \      /       2   \|
  /       2   \ |           \1 + tan (x)/    2*\1 + tan (x)/|
4*\1 + tan (x)/*|2*tan(x) + -------------- - ---------------|
                |                 3               tan(x)    |
                \              tan (x)                      /
4(tan2(x)+1)((tan2(x)+1)2tan3(x)2(tan2(x)+1)tan(x)+2tan(x))4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 2 \tan{\left(x \right)}\right)
The graph
Derivative of y=ln(tgx^2)