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

Derivative of ln(tg^3x)

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

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The solution

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

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

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

    2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    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:

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

      The result of the chain rule is:

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

    The result of the chain rule is:

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

  4. Now simplify:

    6sin⁡(2x)\frac{6}{\sin{\left(2 x \right)}}

The answer is:

6sin⁡(2x)\frac{6}{\sin{\left(2 x \right)}}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
         2   
3 + 3*tan (x)
-------------
    tan(x)
3tan⁡2(x)+3tan⁡(x)\frac{3 \tan^{2}{\left(x \right)} + 3}{\tan{\left(x \right)}}
Simplify
The second derivative [src] 
  /                             2\
  |                /       2   \ |
  |         2      \1 + tan (x)/ |
3*|2 + 2*tan (x) - --------------|
  |                      2       |
  \                   tan (x)    /
3(−(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2)3 \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right)
Simplify
The third derivative [src] 
                /                        2                  \
                |           /       2   \      /       2   \|
  /       2   \ |           \1 + tan (x)/    2*\1 + tan (x)/|
6*\1 + tan (x)/*|2*tan(x) + -------------- - ---------------|
                |                 3               tan(x)    |
                \              tan (x)                      /
6(tan⁡2(x)+1)((tan⁡2(x)+1)2tan⁡3(x)−2(tan⁡2(x)+1)tan⁡(x)+2tan⁡(x))6 \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)
Simplify
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