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Derivada de la función/ sqrt(x)/ 1/e^x/ y=sin/ y=sin(sqrt(x))^(1/e^x)

Derivada de y=sin(sqrt(x))^(1/e^x)

Función f() - derivada -er orden en el punto
v

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
            1 
            --
             x
            E 
/   /  ___\\  
\sin\\/ x //
sin⁡1ex(x)\sin^{\frac{1}{e^{x}}}{\left(\sqrt{x} \right)} 
sin(sqrt(x))^(1/(E^x))
Solución detallada

  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada

    (log⁡(1ex)+1)e−xe−x\left(\log{\left(\frac{1}{e^{x}} \right)} + 1\right) e^{- x e^{- x}}

  2. Simplificamos:

    (1−x)e−xe−x\left(1 - x\right) e^{- x e^{- x}}

Respuesta:

(1−x)e−xe−x\left(1 - x\right) e^{- x e^{- x}}

Primera derivada [src] 
            1                                              
            --                                             
             x                                             
            E  /                             /  ___\  -x  \
/   /  ___\\   |   -x    /   /  ___\\     cos\\/ x /*e    |
\sin\\/ x //  *|- e  *log\sin\\/ x // + ------------------|
               |                            ___    /  ___\|
               \                        2*\/ x *sin\\/ x //
(−e−xlog⁡(sin⁡(x))+e−xcos⁡(x)2xsin⁡(x))sin⁡1ex(x)\left(- e^{- x} \log{\left(\sin{\left(\sqrt{x} \right)} \right)} + \frac{e^{- x} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \sin{\left(\sqrt{x} \right)}}\right) \sin^{\frac{1}{e^{x}}}{\left(\sqrt{x} \right)}
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Segunda derivada [src] 
                  /                                              2                                                                               \    
                  |        /                          /  ___\   \                                                                                |    
                  |        |     /   /  ___\\      cos\\/ x /   |   -x                                                                           |    
            / -x\ |        |2*log\sin\\/ x // - ----------------| *e                                                                             |    
            \e  / |        |                      ___    /  ___\|              /  ___\           2/  ___\            /  ___\                     |    
/   /  ___\\      |   1    \                    \/ x *sin\\/ x //           cos\\/ x /        cos \\/ x /         cos\\/ x /         /   /  ___\\|  -x
\sin\\/ x //     *|- --- + ------------------------------------------- - ---------------- - --------------- - ----------------- + log\sin\\/ x //|*e  
                  |  4*x                        4                          ___    /  ___\          2/  ___\      3/2    /  ___\                  |    
                  \                                                      \/ x *sin\\/ x /   4*x*sin \\/ x /   4*x   *sin\\/ x /                  /
((2log⁡(sin⁡(x))−cos⁡(x)xsin⁡(x))2e−x4+log⁡(sin⁡(x))−14x−cos⁡2(x)4xsin⁡2(x)−cos⁡(x)xsin⁡(x)−cos⁡(x)4x32sin⁡(x))e−xsin⁡e−x(x)\left(\frac{\left(2 \log{\left(\sin{\left(\sqrt{x} \right)} \right)} - \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \sin{\left(\sqrt{x} \right)}}\right)^{2} e^{- x}}{4} + \log{\left(\sin{\left(\sqrt{x} \right)} \right)} - \frac{1}{4 x} - \frac{\cos^{2}{\left(\sqrt{x} \right)}}{4 x \sin^{2}{\left(\sqrt{x} \right)}} - \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \sin{\left(\sqrt{x} \right)}} - \frac{\cos{\left(\sqrt{x} \right)}}{4 x^{\frac{3}{2}} \sin{\left(\sqrt{x} \right)}}\right) e^{- x} \sin^{e^{- x}}{\left(\sqrt{x} \right)}
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Tercera derivada [src] 
                  /                                                                       3                                                                                                                                                                                                                                                       \    
                  |                                 /                          /  ___\   \                                                                                                                                 /                          /  ___\   \ /                            2/  ___\          /  ___\            /  ___\  \    |    
                  |                                 |     /   /  ___\\      cos\\/ x /   |   -2*x                                                                                                                          |     /   /  ___\\      cos\\/ x /   | |1        /   /  ___\\    cos \\/ x /       cos\\/ x /       4*cos\\/ x /  |  -x|    
            / -x\ |                                 |2*log\sin\\/ x // - ----------------| *e                                                                                                                            3*|2*log\sin\\/ x // - ----------------|*|- - 4*log\sin\\/ x // + ------------- + --------------- + ----------------|*e  |    
            \e  / |                                 |                      ___    /  ___\|                /  ___\           3/  ___\               /  ___\            2/  ___\          2/  ___\             /  ___\       |                      ___    /  ___\| |x                            2/  ___\    3/2    /  ___\     ___    /  ___\|    |    
/   /  ___\\      |     /   /  ___\\    3     3     \                    \/ x *sin\\/ x //             cos\\/ x /        cos \\/ x /          3*cos\\/ x /       3*cos \\/ x /     3*cos \\/ x /        3*cos\\/ x /       \                    \/ x *sin\\/ x // \                        x*sin \\/ x /   x   *sin\\/ x /   \/ x *sin\\/ x //    |  -x
\sin\\/ x //     *|- log\sin\\/ x // + --- + ---- - --------------------------------------------- + --------------- + ------------------ + ------------------ + --------------- + ---------------- + ----------------- + -------------------------------------------------------------------------------------------------------------------------|*e  
                  |                    4*x      2                         8                          3/2    /  ___\      3/2    3/  ___\       ___    /  ___\          2/  ___\      2    2/  ___\      5/2    /  ___\                                                               8                                                            |    
                  \                          8*x                                                    x   *sin\\/ x /   4*x   *sin \\/ x /   2*\/ x *sin\\/ x /   4*x*sin \\/ x /   8*x *sin \\/ x /   8*x   *sin\\/ x /                                                                                                                            /
(−(2log⁡(sin⁡(x))−cos⁡(x)xsin⁡(x))3e−2x8+3(2log⁡(sin⁡(x))−cos⁡(x)xsin⁡(x))(−4log⁡(sin⁡(x))+1x+cos⁡2(x)xsin⁡2(x)+4cos⁡(x)xsin⁡(x)+cos⁡(x)x32sin⁡(x))e−x8−log⁡(sin⁡(x))+34x+3cos⁡2(x)4xsin⁡2(x)+38x2+3cos⁡2(x)8x2sin⁡2(x)+3cos⁡(x)2xsin⁡(x)+cos⁡(x)x32sin⁡(x)+cos⁡3(x)4x32sin⁡3(x)+3cos⁡(x)8x52sin⁡(x))e−xsin⁡e−x(x)\left(- \frac{\left(2 \log{\left(\sin{\left(\sqrt{x} \right)} \right)} - \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \sin{\left(\sqrt{x} \right)}}\right)^{3} e^{- 2 x}}{8} + \frac{3 \left(2 \log{\left(\sin{\left(\sqrt{x} \right)} \right)} - \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \sin{\left(\sqrt{x} \right)}}\right) \left(- 4 \log{\left(\sin{\left(\sqrt{x} \right)} \right)} + \frac{1}{x} + \frac{\cos^{2}{\left(\sqrt{x} \right)}}{x \sin^{2}{\left(\sqrt{x} \right)}} + \frac{4 \cos{\left(\sqrt{x} \right)}}{\sqrt{x} \sin{\left(\sqrt{x} \right)}} + \frac{\cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}} \sin{\left(\sqrt{x} \right)}}\right) e^{- x}}{8} - \log{\left(\sin{\left(\sqrt{x} \right)} \right)} + \frac{3}{4 x} + \frac{3 \cos^{2}{\left(\sqrt{x} \right)}}{4 x \sin^{2}{\left(\sqrt{x} \right)}} + \frac{3}{8 x^{2}} + \frac{3 \cos^{2}{\left(\sqrt{x} \right)}}{8 x^{2} \sin^{2}{\left(\sqrt{x} \right)}} + \frac{3 \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \sin{\left(\sqrt{x} \right)}} + \frac{\cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}} \sin{\left(\sqrt{x} \right)}} + \frac{\cos^{3}{\left(\sqrt{x} \right)}}{4 x^{\frac{3}{2}} \sin^{3}{\left(\sqrt{x} \right)}} + \frac{3 \cos{\left(\sqrt{x} \right)}}{8 x^{\frac{5}{2}} \sin{\left(\sqrt{x} \right)}}\right) e^{- x} \sin^{e^{- x}}{\left(\sqrt{x} \right)}
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