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ln(tgx/2)
The derivative of the function/ ln(tgx/2)

Derivative of ln(tgx/2)

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

You have entered [src] 
   /tan(x)\
log|------|
   \  2   /
log⁡(tan⁡(x)2)\log{\left(\frac{\tan{\left(x \right)}}{2} \right)} 
d /   /tan(x)\\
--|log|------||
dx\   \  2   //
ddxlog⁡(tan⁡(x)2)\frac{d}{d x} \log{\left(\frac{\tan{\left(x \right)}}{2} \right)}
Transform
Detail solution

  1. Let u=tan⁡(x)2u = \frac{\tan{\left(x \right)}}{2}.

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

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

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      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)}}

      So, the result is: sin⁡2(x)+cos⁡2(x)2cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)}}

    The result of the chain rule is:

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

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-200200
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The first derivative [src] 
  /       2   \
  |1   tan (x)|
2*|- + -------|
  \2      2   /
---------------
     tan(x)
2(tan⁡2(x)2+12)tan⁡(x)\frac{2 \left(\frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)}{\tan{\left(x \right)}}
Simplify
The second derivative [src] 
                             2
                /       2   \ 
         2      \1 + tan (x)/ 
2 + 2*tan (x) - --------------
                      2       
                   tan (x)
2tan⁡2(x)−(tan⁡2(x)+1)2tan⁡2(x)+22 \tan^{2}{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2
Simplify
The third derivative [src] 
                /                        2                  \
                |           /       2   \      /       2   \|
  /       2   \ |           \1 + tan (x)/    2*\1 + tan (x)/|
2*\1 + tan (x)/*|2*tan(x) + -------------- - ---------------|
                |                 3               tan(x)    |
                \              tan (x)                      /
2(tan⁡2(x)+1)(2tan⁡(x)−2(tan⁡2(x)+1)tan⁡(x)+(tan⁡2(x)+1)2tan⁡3(x))2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}}\right)
Simplify
The graph
Derivative of ln(tgx/2)
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