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y=ex^√sinx
Derivada de la función/ √sinx/ y=ex^√sinx

Derivada de y=ex^√sinx

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

Gráfico:

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Solución

Ha introducido [src] 
 /   ________\
 | \/ sin(x) |
 \x          /
E
exsin⁡(x)e^{x^{\sqrt{\sin{\left(x \right)}}}} 
E^(x^(sqrt(sin(x))))
Solución detallada

  1. Sustituimos u=xsin⁡(x)u = x^{\sqrt{\sin{\left(x \right)}}}.

  2. Derivado eue^{u} es.

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxxsin⁡(x)\frac{d}{d x} x^{\sqrt{\sin{\left(x \right)}}}:

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

      Perola derivada

      (log⁡(sin⁡(x))+1)sin⁡sin⁡(x)2(x)\left(\log{\left(\sqrt{\sin{\left(x \right)}} \right)} + 1\right) \sin^{\frac{\sqrt{\sin{\left(x \right)}}}{2}}{\left(x \right)}

    Como resultado de la secuencia de reglas:

    (log⁡(sin⁡(x))+1)exsin⁡(x)sin⁡sin⁡(x)2(x)\left(\log{\left(\sqrt{\sin{\left(x \right)}} \right)} + 1\right) e^{x^{\sqrt{\sin{\left(x \right)}}}} \sin^{\frac{\sqrt{\sin{\left(x \right)}}}{2}}{\left(x \right)}

  4. Simplificamos:

    (log⁡(sin⁡(x))+2)exsin⁡(x)sin⁡sin⁡(x)2(x)2\frac{\left(\log{\left(\sin{\left(x \right)} \right)} + 2\right) e^{x^{\sqrt{\sin{\left(x \right)}}}} \sin^{\frac{\sqrt{\sin{\left(x \right)}}}{2}}{\left(x \right)}}{2}

Respuesta:

(log⁡(sin⁡(x))+2)exsin⁡(x)sin⁡sin⁡(x)2(x)2\frac{\left(\log{\left(\sin{\left(x \right)} \right)} + 2\right) e^{x^{\sqrt{\sin{\left(x \right)}}}} \sin^{\frac{\sqrt{\sin{\left(x \right)}}}{2}}{\left(x \right)}}{2}

Gráfica
02468-8-6-4-2-1010-1000010000
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Trazar el gráfico f'(x)
Primera derivada [src] 
                                          /   ________\
   ________ /  ________                \  | \/ sin(x) |
 \/ sin(x)  |\/ sin(x)    cos(x)*log(x)|  \x          /
x          *|---------- + -------------|*e             
            |    x             ________|               
            \              2*\/ sin(x) /
xsin⁡(x)(log⁡(x)cos⁡(x)2sin⁡(x)+sin⁡(x)x)exsin⁡(x)x^{\sqrt{\sin{\left(x \right)}}} \left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} + \frac{\sqrt{\sin{\left(x \right)}}}{x}\right) e^{x^{\sqrt{\sin{\left(x \right)}}}}
Simplificar
Segunda derivada [src] 
            /                              2                                                                              2                                \               
            |/    ________                \                                        ________ /    ________                \                                 |               
            ||2*\/ sin(x)    cos(x)*log(x)|                                      \/ sin(x)  |2*\/ sin(x)    cos(x)*log(x)|                                 |               
            ||------------ + -------------|                                     x          *|------------ + -------------|                                 |  /   ________\
   ________ ||     x             ________ |      ________     ________                      |     x             ________ |                      2          |  | \/ sin(x) |
 \/ sin(x)  |\                 \/ sin(x)  /    \/ sin(x)    \/ sin(x) *log(x)               \                 \/ sin(x)  /       cos(x)      cos (x)*log(x)|  \x          /
x          *|------------------------------- - ---------- - ----------------- + ------------------------------------------- + ------------ - --------------|*e             
            |               4                       2               2                                4                            ________         3/2     |               
            \                                      x                                                                          x*\/ sin(x)     4*sin   (x)  /
xsin⁡(x)(xsin⁡(x)(log⁡(x)cos⁡(x)sin⁡(x)+2sin⁡(x)x)24+(log⁡(x)cos⁡(x)sin⁡(x)+2sin⁡(x)x)24−log⁡(x)sin⁡(x)2−log⁡(x)cos⁡2(x)4sin⁡32(x)+cos⁡(x)xsin⁡(x)−sin⁡(x)x2)exsin⁡(x)x^{\sqrt{\sin{\left(x \right)}}} \left(\frac{x^{\sqrt{\sin{\left(x \right)}}} \left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x}\right)^{2}}{4} + \frac{\left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x}\right)^{2}}{4} - \frac{\log{\left(x \right)} \sqrt{\sin{\left(x \right)}}}{2} - \frac{\log{\left(x \right)} \cos^{2}{\left(x \right)}}{4 \sin^{\frac{3}{2}}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{x \sqrt{\sin{\left(x \right)}}} - \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\right) e^{x^{\sqrt{\sin{\left(x \right)}}}}
Simplificar
Tercera derivada [src] 
            /                              3                                                                                                                                                                                     3                                               3                                                                                                                                                                                         \               
            |/    ________                \                                    /    ________                \ /                          ________      2                         \        ________ /    ________                \         ________ /    ________                \                                           ________ /    ________                \ /                          ________      2                         \                                   |               
            ||2*\/ sin(x)    cos(x)*log(x)|                                    |2*\/ sin(x)    cos(x)*log(x)| |    ________          4*\/ sin(x)    cos (x)*log(x)     4*cos(x)  |    2*\/ sin(x)  |2*\/ sin(x)    cos(x)*log(x)|       \/ sin(x)  |2*\/ sin(x)    cos(x)*log(x)|                                         \/ sin(x)  |2*\/ sin(x)    cos(x)*log(x)| |    ________          4*\/ sin(x)    cos (x)*log(x)     4*cos(x)  |                                   |               
            ||------------ + -------------|                                  3*|------------ + -------------|*|2*\/ sin(x) *log(x) + ------------ + -------------- - ------------|   x            *|------------ + -------------|    3*x          *|------------ + -------------|                                      3*x          *|------------ + -------------|*|2*\/ sin(x) *log(x) + ------------ + -------------- - ------------|                                   |  /   ________\
   ________ ||     x             ________ |        ________       ________     |     x             ________ | |                            2             3/2             ________|                 |     x             ________ |                  |     x             ________ |                             2                      |     x             ________ | |                            2             3/2             ________|                        3          |  | \/ sin(x) |
 \/ sin(x)  |\                 \/ sin(x)  /    2*\/ sin(x)    3*\/ sin(x)      \                 \/ sin(x)  / \                           x           sin   (x)      x*\/ sin(x) /                 \                 \/ sin(x)  /                  \                 \/ sin(x)  /        3*cos(x)        3*cos (x)                   \                 \/ sin(x)  / \                           x           sin   (x)      x*\/ sin(x) /   cos(x)*log(x)   3*cos (x)*log(x)|  \x          /
x          *|------------------------------- + ------------ - ------------ - ----------------------------------------------------------------------------------------------------- + --------------------------------------------- + --------------------------------------------- - --------------- - ------------- - ----------------------------------------------------------------------------------------------------------------- + ------------- + ----------------|*e             
            |               8                        3            2*x                                                          8                                                                           8                                               8                            2   ________          3/2                                                              8                                                                ________          5/2      |               
            \                                       x                                                                                                                                                                                                                                2*x *\/ sin(x)    4*x*sin   (x)                                                                                                                        4*\/ sin(x)      8*sin   (x)   /
xsin⁡(x)(x2sin⁡(x)(log⁡(x)cos⁡(x)sin⁡(x)+2sin⁡(x)x)38+3xsin⁡(x)(log⁡(x)cos⁡(x)sin⁡(x)+2sin⁡(x)x)38−3xsin⁡(x)(log⁡(x)cos⁡(x)sin⁡(x)+2sin⁡(x)x)(2log⁡(x)sin⁡(x)+log⁡(x)cos⁡2(x)sin⁡32(x)−4cos⁡(x)xsin⁡(x)+4sin⁡(x)x2)8+(log⁡(x)cos⁡(x)sin⁡(x)+2sin⁡(x)x)38−3(log⁡(x)cos⁡(x)sin⁡(x)+2sin⁡(x)x)(2log⁡(x)sin⁡(x)+log⁡(x)cos⁡2(x)sin⁡32(x)−4cos⁡(x)xsin⁡(x)+4sin⁡(x)x2)8+log⁡(x)cos⁡(x)4sin⁡(x)+3log⁡(x)cos⁡3(x)8sin⁡52(x)−3sin⁡(x)2x−3cos⁡2(x)4xsin⁡32(x)−3cos⁡(x)2x2sin⁡(x)+2sin⁡(x)x3)exsin⁡(x)x^{\sqrt{\sin{\left(x \right)}}} \left(\frac{x^{2 \sqrt{\sin{\left(x \right)}}} \left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x}\right)^{3}}{8} + \frac{3 x^{\sqrt{\sin{\left(x \right)}}} \left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x}\right)^{3}}{8} - \frac{3 x^{\sqrt{\sin{\left(x \right)}}} \left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x}\right) \left(2 \log{\left(x \right)} \sqrt{\sin{\left(x \right)}} + \frac{\log{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}} - \frac{4 \cos{\left(x \right)}}{x \sqrt{\sin{\left(x \right)}}} + \frac{4 \sqrt{\sin{\left(x \right)}}}{x^{2}}\right)}{8} + \frac{\left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x}\right)^{3}}{8} - \frac{3 \left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x}\right) \left(2 \log{\left(x \right)} \sqrt{\sin{\left(x \right)}} + \frac{\log{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}} - \frac{4 \cos{\left(x \right)}}{x \sqrt{\sin{\left(x \right)}}} + \frac{4 \sqrt{\sin{\left(x \right)}}}{x^{2}}\right)}{8} + \frac{\log{\left(x \right)} \cos{\left(x \right)}}{4 \sqrt{\sin{\left(x \right)}}} + \frac{3 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{8 \sin^{\frac{5}{2}}{\left(x \right)}} - \frac{3 \sqrt{\sin{\left(x \right)}}}{2 x} - \frac{3 \cos^{2}{\left(x \right)}}{4 x \sin^{\frac{3}{2}}{\left(x \right)}} - \frac{3 \cos{\left(x \right)}}{2 x^{2} \sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x^{3}}\right) e^{x^{\sqrt{\sin{\left(x \right)}}}}
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Gráfico
Derivada de y=ex^√sinx
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