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log(tan(x))
The derivative of the function/ log(tan(x))

Derivative of log(tan(x))

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

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

  1. Let u=tan⁡(x)u = \tan{\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⁡(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:

    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
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       2   
1 + tan (x)
-----------
   tan(x)
tan⁡2(x)+1tan⁡(x)\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}}
Simplify
The second derivative [src] 
                             2
                /       2   \ 
         2      \1 + tan (x)/ 
2 + 2*tan (x) - --------------
                      2       
                   tan (x)
−(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \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)((tan⁡2(x)+1)2tan⁡3(x)−2(tan⁡2(x)+1)tan⁡(x)+2tan⁡(x))2 \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
The graph
Derivative of log(tan(x))
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