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y=lntg^2x
The derivative of the function/ y=lntg^2x

Derivative of y=lntg^2x

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

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The solution

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   2        
log (tan(x))
log⁡(tan⁡(x))2\log{\left(\tan{\left(x \right)} \right)}^{2} 
log(tan(x))^2
Detail solution

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

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

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

    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)}}

    The result of the chain rule is:

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

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  /       2   \            
2*\1 + tan (x)/*log(tan(x))
---------------------------
           tan(x)
2(tan⁡2(x)+1)log⁡(tan⁡(x))tan⁡(x)\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}}{\tan{\left(x \right)}}
Simplify
The second derivative [src] 
                /                       2      /       2   \            \
  /       2   \ |                1 + tan (x)   \1 + tan (x)/*log(tan(x))|
2*\1 + tan (x)/*|2*log(tan(x)) + ----------- - -------------------------|
                |                     2                    2            |
                \                  tan (x)              tan (x)         /
2(tan⁡2(x)+1)(−(tan⁡2(x)+1)log⁡(tan⁡(x))tan⁡2(x)+tan⁡2(x)+1tan⁡2(x)+2log⁡(tan⁡(x)))2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}}{\tan^{2}{\left(x \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} + 2 \log{\left(\tan{\left(x \right)} \right)}\right)
Simplify
The third derivative [src] 
                /                 2                                                                                         2            \
                |    /       2   \                             /       2   \     /       2   \                 /       2   \             |
  /       2   \ |  3*\1 + tan (x)/                           6*\1 + tan (x)/   4*\1 + tan (x)/*log(tan(x))   2*\1 + tan (x)/ *log(tan(x))|
2*\1 + tan (x)/*|- ---------------- + 4*log(tan(x))*tan(x) + --------------- - --------------------------- + ----------------------------|
                |         3                                       tan(x)                  tan(x)                          3              |
                \      tan (x)                                                                                         tan (x)           /
2(tan⁡2(x)+1)(2(tan⁡2(x)+1)2log⁡(tan⁡(x))tan⁡3(x)−3(tan⁡2(x)+1)2tan⁡3(x)−4(tan⁡2(x)+1)log⁡(tan⁡(x))tan⁡(x)+6(tan⁡2(x)+1)tan⁡(x)+4log⁡(tan⁡(x))tan⁡(x))2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)}}{\tan^{3}{\left(x \right)}} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}}{\tan{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 4 \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right)
Simplify
The graph
Derivative of y=lntg^2x
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