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The derivative of the function/ (sqrt(tan(x/(e^x))))

Derivative of (sqrt(tan(x/(e^x))))

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

You have entered [src] 
     _________
    /    /x \ 
   /  tan|--| 
  /      | x| 
\/       \E /
tan⁡(xex)\sqrt{\tan{\left(\frac{x}{e^{x}} \right)}} 
sqrt(tan(x/E^x))
Detail solution

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

  2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

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

    1. Rewrite the function to be differentiated:

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

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

      1. Let u=xexu = \frac{x}{e^{x}}.

      2. The derivative of sine is cosine:

        ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddxxex\frac{d}{d x} \frac{x}{e^{x}}:

        1. 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)=xf{\left(x \right)} = x and g(x)=exg{\left(x \right)} = e^{x}.

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

          1. Apply the power rule: xx goes to 11

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

          1. The derivative of exe^{x} is itself.

          Now plug in to the quotient rule:

          (−xex+ex)e−2x\left(- x e^{x} + e^{x}\right) e^{- 2 x}

        The result of the chain rule is:

        (−xex+ex)e−2xcos⁡(xex)\left(- x e^{x} + e^{x}\right) e^{- 2 x} \cos{\left(\frac{x}{e^{x}} \right)}

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

      1. Let u=xexu = \frac{x}{e^{x}}.

      2. The derivative of cosine is negative sine:

        dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddxxex\frac{d}{d x} \frac{x}{e^{x}}:

        1. 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)=xf{\left(x \right)} = x and g(x)=exg{\left(x \right)} = e^{x}.

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

          1. Apply the power rule: xx goes to 11

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

          1. The derivative of exe^{x} is itself.

          Now plug in to the quotient rule:

          (−xex+ex)e−2x\left(- x e^{x} + e^{x}\right) e^{- 2 x}

        The result of the chain rule is:

        −(−xex+ex)e−2xsin⁡(xex)- \left(- x e^{x} + e^{x}\right) e^{- 2 x} \sin{\left(\frac{x}{e^{x}} \right)}

      Now plug in to the quotient rule:

      (−xex+ex)e−2xsin⁡2(xex)+(−xex+ex)e−2xcos⁡2(xex)cos⁡2(xex)\frac{\left(- x e^{x} + e^{x}\right) e^{- 2 x} \sin^{2}{\left(\frac{x}{e^{x}} \right)} + \left(- x e^{x} + e^{x}\right) e^{- 2 x} \cos^{2}{\left(\frac{x}{e^{x}} \right)}}{\cos^{2}{\left(\frac{x}{e^{x}} \right)}}

    The result of the chain rule is:

    (−xex+ex)e−2xsin⁡2(xex)+(−xex+ex)e−2xcos⁡2(xex)2cos⁡2(xex)tan⁡(xex)\frac{\left(- x e^{x} + e^{x}\right) e^{- 2 x} \sin^{2}{\left(\frac{x}{e^{x}} \right)} + \left(- x e^{x} + e^{x}\right) e^{- 2 x} \cos^{2}{\left(\frac{x}{e^{x}} \right)}}{2 \cos^{2}{\left(\frac{x}{e^{x}} \right)} \sqrt{\tan{\left(\frac{x}{e^{x}} \right)}}}

  4. Now simplify:

    (1−x)e−x2cos⁡2(xe−x)tan⁡(xe−x)\frac{\left(1 - x\right) e^{- x}}{2 \cos^{2}{\left(x e^{- x} \right)} \sqrt{\tan{\left(x e^{- x} \right)}}}

The answer is:

(1−x)e−x2cos⁡2(xe−x)tan⁡(xe−x)\frac{\left(1 - x\right) e^{- x}}{2 \cos^{2}{\left(x e^{- x} \right)} \sqrt{\tan{\left(x e^{- x} \right)}}}

The graph
02468-8-6-4-2-1010-20000002000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/       2/x \\ /1       -x\
|1 + tan |--||*|-- - x*e  |
|        | x|| | x        |
\        \E // \E         /
---------------------------
             _________     
            /    /x \      
      2*   /  tan|--|      
          /      | x|      
        \/       \E /
(−xe−x+1ex)(tan⁡2(xex)+1)2tan⁡(xex)\frac{\left(- x e^{- x} + \frac{1}{e^{x}}\right) \left(\tan^{2}{\left(\frac{x}{e^{x}} \right)} + 1\right)}{2 \sqrt{\tan{\left(\frac{x}{e^{x}} \right)}}}
Simplify
The second derivative [src] 
                  /                                 ____________               2 /       2/   -x\\  -x\    
/       2/   -x\\ |      -2 + x                2   /    /   -x\   -x   (-1 + x) *\1 + tan \x*e  //*e  |  -x
\1 + tan \x*e  //*|----------------- + (-1 + x) *\/  tan\x*e  / *e   - -------------------------------|*e  
                  |     ____________                                                3/2/   -x\        |    
                  |    /    /   -x\                                            4*tan   \x*e  /        |    
                  \2*\/  tan\x*e  /                                                                   /
(tan⁡2(xe−x)+1)(x−22tan⁡(xe−x)−(x−1)2(tan⁡2(xe−x)+1)e−x4tan⁡32(xe−x)+(x−1)2e−xtan⁡(xe−x))e−x\left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) \left(\frac{x - 2}{2 \sqrt{\tan{\left(x e^{- x} \right)}}} - \frac{\left(x - 1\right)^{2} \left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) e^{- x}}{4 \tan^{\frac{3}{2}}{\left(x e^{- x} \right)}} + \left(x - 1\right)^{2} e^{- x} \sqrt{\tan{\left(x e^{- x} \right)}}\right) e^{- x}
Simplify
The third derivative [src] 
                  /                                                                                                                                                         2                                                            \    
                  |                                                                3 /       2/   -x\\  -2*x        ____________                           /       2/   -x\\          3  -2*x     /       2/   -x\\                    -x|    
/       2/   -x\\ |        -3 + x                  3    3/2/   -x\  -2*x   (-1 + x) *\1 + tan \x*e  //*e           /    /   -x\                     -x   3*\1 + tan \x*e  // *(-1 + x) *e       3*\1 + tan \x*e  //*(-1 + x)*(-2 + x)*e  |  -x
\1 + tan \x*e  //*|- ----------------- - 2*(-1 + x) *tan   \x*e  /*e     + --------------------------------- - 3*\/  tan\x*e  / *(-1 + x)*(-2 + x)*e   - ------------------------------------ + -----------------------------------------|*e  
                  |       ____________                                                  ____________                                                                    5/2/   -x\                                3/2/   -x\             |    
                  |      /    /   -x\                                                  /    /   -x\                                                                8*tan   \x*e  /                           4*tan   \x*e  /             |    
                  \  2*\/  tan\x*e  /                                              2*\/  tan\x*e  /                                                                                                                                      /
(tan⁡2(xe−x)+1)(−x−32tan⁡(xe−x)+3(x−2)(x−1)(tan⁡2(xe−x)+1)e−x4tan⁡32(xe−x)−3(x−2)(x−1)e−xtan⁡(xe−x)−3(x−1)3(tan⁡2(xe−x)+1)2e−2x8tan⁡52(xe−x)+(x−1)3(tan⁡2(xe−x)+1)e−2x2tan⁡(xe−x)−2(x−1)3e−2xtan⁡32(xe−x))e−x\left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) \left(- \frac{x - 3}{2 \sqrt{\tan{\left(x e^{- x} \right)}}} + \frac{3 \left(x - 2\right) \left(x - 1\right) \left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) e^{- x}}{4 \tan^{\frac{3}{2}}{\left(x e^{- x} \right)}} - 3 \left(x - 2\right) \left(x - 1\right) e^{- x} \sqrt{\tan{\left(x e^{- x} \right)}} - \frac{3 \left(x - 1\right)^{3} \left(\tan^{2}{\left(x e^{- x} \right)} + 1\right)^{2} e^{- 2 x}}{8 \tan^{\frac{5}{2}}{\left(x e^{- x} \right)}} + \frac{\left(x - 1\right)^{3} \left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) e^{- 2 x}}{2 \sqrt{\tan{\left(x e^{- x} \right)}}} - 2 \left(x - 1\right)^{3} e^{- 2 x} \tan^{\frac{3}{2}}{\left(x e^{- x} \right)}\right) e^{- x}
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