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x^sinxln(x+(1+x^2)^1/2)
Derivada de la función/ 1+x^2/ x^sin/ xln(x+(1+x^2)^1/2)/ x^sinxln(x+(1+x^2)^1/2)

Derivada de x^sinxln(x+(1+x^2)^1/2)

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src] 
           /       ________\
 sin(x)    |      /      2 |
x      *log\x + \/  1 + x  /
xsin⁡(x)log⁡(x+x2+1)x^{\sin{\left(x \right)}} \log{\left(x + \sqrt{x^{2} + 1} \right)} 
x^sin(x)*log(x + sqrt(1 + x^2))
Solución detallada

  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=xsin⁡(x)f{\left(x \right)} = x^{\sin{\left(x \right)}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      Perola derivada

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

    g(x)=log⁡(x+x2+1)g{\left(x \right)} = \log{\left(x + \sqrt{x^{2} + 1} \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Sustituimos u=x+x2+1u = x + \sqrt{x^{2} + 1}.

    2. Derivado log⁡(u)\log{\left(u \right)} es 1u\frac{1}{u}.

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddx(x+x2+1)\frac{d}{d x} \left(x + \sqrt{x^{2} + 1}\right):

      1. diferenciamos x+x2+1x + \sqrt{x^{2} + 1} miembro por miembro:

        1. Según el principio, aplicamos: xx tenemos 11

        2. Sustituimos u=x2+1u = x^{2} + 1.

        3. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

        4. Luego se aplica una cadena de reglas. Multiplicamos por ddx(x2+1)\frac{d}{d x} \left(x^{2} + 1\right):

          1. diferenciamos x2+1x^{2} + 1 miembro por miembro:

            1. La derivada de una constante 11 es igual a cero.

            2. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

            Como resultado de: 2x2 x

          Como resultado de la secuencia de reglas:

          xx2+1\frac{x}{\sqrt{x^{2} + 1}}

        Como resultado de: xx2+1+1\frac{x}{\sqrt{x^{2} + 1}} + 1

      Como resultado de la secuencia de reglas:

      xx2+1+1x+x2+1\frac{\frac{x}{\sqrt{x^{2} + 1}} + 1}{x + \sqrt{x^{2} + 1}}

    Como resultado de: xsin⁡(x)(xx2+1+1)x+x2+1+(log⁡(sin⁡(x))+1)log⁡(x+x2+1)sin⁡sin⁡(x)(x)\frac{x^{\sin{\left(x \right)}} \left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)}{x + \sqrt{x^{2} + 1}} + \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x + \sqrt{x^{2} + 1} \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)}

  2. Simplificamos:

    xsin⁡(x)+x2+1(log⁡(sin⁡(x))+1)log⁡(x+x2+1)sin⁡sin⁡(x)(x)x2+1\frac{x^{\sin{\left(x \right)}} + \sqrt{x^{2} + 1} \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x + \sqrt{x^{2} + 1} \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)}}{\sqrt{x^{2} + 1}}

Respuesta:

xsin⁡(x)+x2+1(log⁡(sin⁡(x))+1)log⁡(x+x2+1)sin⁡sin⁡(x)(x)x2+1\frac{x^{\sin{\left(x \right)}} + \sqrt{x^{2} + 1} \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x + \sqrt{x^{2} + 1} \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)}}{\sqrt{x^{2} + 1}}

Gráfica
02468-8-6-4-2-1010-5050
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
 sin(x) /         x     \                                                        
x      *|1 + -----------|                                                        
        |       ________|                                                        
        |      /      2 |                                       /       ________\
        \    \/  1 + x  /    sin(x) /sin(x)                \    |      /      2 |
------------------------- + x      *|------ + cos(x)*log(x)|*log\x + \/  1 + x  /
            ________                \  x                   /                     
           /      2                                                              
     x + \/  1 + x
xsin⁡(x)(log⁡(x)cos⁡(x)+sin⁡(x)x)log⁡(x+x2+1)+xsin⁡(x)(xx2+1+1)x+x2+1x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \log{\left(x + \sqrt{x^{2} + 1} \right)} + \frac{x^{\sin{\left(x \right)}} \left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)}{x + \sqrt{x^{2} + 1}}
Simplificar
Segunda derivada [src] 
        /                                                                                                                      2                                               \
        |                                                                                               2     /         x     \                                                |
        |                                                                                              x      |1 + -----------|                                                |
        |                                                                                       -1 + ------   |       ________|                                                |
        |                                                                                                 2   |      /      2 |                                                |
        |                                                                                            1 + x    \    \/  1 + x  /      /         x     \ /sin(x)                \|
        |                                                                                       ----------- + ------------------   2*|1 + -----------|*|------ + cos(x)*log(x)||
        |                                                                                          ________           ________       |       ________| \  x                   /|
        |/                        2                                    \    /       ________\     /      2           /      2        |      /      2 |                         |
 sin(x) ||/sin(x)                \    sin(x)                   2*cos(x)|    |      /      2 |   \/  1 + x      x + \/  1 + x         \    \/  1 + x  /                         |
x      *|||------ + cos(x)*log(x)|  - ------ - log(x)*sin(x) + --------|*log\x + \/  1 + x  / - -------------------------------- + --------------------------------------------|
        ||\  x                   /       2                        x    |                                       ________                                 ________               |
        |\                              x                              /                                      /      2                                 /      2                |
        \                                                                                               x + \/  1 + x                            x + \/  1 + x                 /
xsin⁡(x)(((log⁡(x)cos⁡(x)+sin⁡(x)x)2−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2)log⁡(x+x2+1)+2(xx2+1+1)(log⁡(x)cos⁡(x)+sin⁡(x)x)x+x2+1−x2x2+1−1x2+1+(xx2+1+1)2x+x2+1x+x2+1)x^{\sin{\left(x \right)}} \left(\left(\left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} - \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}\right) \log{\left(x + \sqrt{x^{2} + 1} \right)} + \frac{2 \left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right) \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)}{x + \sqrt{x^{2} + 1}} - \frac{\frac{\frac{x^{2}}{x^{2} + 1} - 1}{\sqrt{x^{2} + 1}} + \frac{\left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)^{2}}{x + \sqrt{x^{2} + 1}}}{x + \sqrt{x^{2} + 1}}\right)
Simplificar
Tercera derivada [src] 
        /                   3                                           /        2  \                                                                                                                                                                                                                                                                                                                             \
        |  /         x     \        /        2  \     /         x     \ |       x   |                                                                                                                                                                                                     /                               2\                                                                                      |
        |2*|1 + -----------|        |       x   |   3*|1 + -----------|*|-1 + ------|                                                                                                                                                                                                     |        2     /         x     \ |                                                                                      |
        |  |       ________|    3*x*|-1 + ------|     |       ________| |          2|                                                                                                                                                                                                     |       x      |1 + -----------| |                                                                                      |
        |  |      /      2 |        |          2|     |      /      2 | \     1 + x /                                                                                                                                                                                                     |-1 + ------   |       ________| |                                                                                      |
        |  \    \/  1 + x  /        \     1 + x /     \    \/  1 + x  /                                                                                                                                                                                                                   |          2   |      /      2 | |                       /                        2                                    \|
        |-------------------- + ----------------- + ---------------------------------                                                                                                                                                                            /sin(x)                \ |     1 + x    \    \/  1 + x  / |     /         x     \ |/sin(x)                \    sin(x)                   2*cos(x)||
        |                  2               3/2           ________ /       ________\                                                                                                                                                                            3*|------ + cos(x)*log(x)|*|----------- + ------------------|   3*|1 + -----------|*||------ + cos(x)*log(x)|  - ------ - log(x)*sin(x) + --------||
        | /       ________\        /     2\             /      2  |      /      2 |                                                                                                                                                                              \  x                   / |   ________           ________  |     |       ________| |\  x                   /       2                        x    ||
        | |      /      2 |        \1 + x /           \/  1 + x  *\x + \/  1 + x  /     /                          3                                                                                                                  \    /       ________\                              |  /      2           /      2   |     |      /      2 | \                              x                              /|
 sin(x) | \x + \/  1 + x  /                                                             |  /sin(x)                \                    2*sin(x)   3*sin(x)   3*cos(x)     /sin(x)                \ /sin(x)                   2*cos(x)\|    |      /      2 |                              \\/  1 + x      x + \/  1 + x    /     \    \/  1 + x  /                                                                |
x      *|---------------------------------------------------------------------------- - |- |------ + cos(x)*log(x)|  + cos(x)*log(x) - -------- + -------- + -------- + 3*|------ + cos(x)*log(x)|*|------ + log(x)*sin(x) - --------||*log\x + \/  1 + x  / - ------------------------------------------------------------- + -----------------------------------------------------------------------------------|
        |                                     ________                                  |  \  x                   /                        3         x           2        \  x                   / |   2                        x    ||                                                      ________                                                                   ________                                  |
        |                                    /      2                                   \                                                 x                     x                                  \  x                              //                                                     /      2                                                                   /      2                                   |
        \                              x + \/  1 + x                                                                                                                                                                                                                                  x + \/  1 + x                                                              x + \/  1 + x                                    /
xsin⁡(x)(−(−(log⁡(x)cos⁡(x)+sin⁡(x)x)3+3(log⁡(x)cos⁡(x)+sin⁡(x)x)(log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)+log⁡(x)cos⁡(x)+3sin⁡(x)x+3cos⁡(x)x2−2sin⁡(x)x3)log⁡(x+x2+1)+3(xx2+1+1)((log⁡(x)cos⁡(x)+sin⁡(x)x)2−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2)x+x2+1−3(x2x2+1−1x2+1+(xx2+1+1)2x+x2+1)(log⁡(x)cos⁡(x)+sin⁡(x)x)x+x2+1+3x(x2x2+1−1)(x2+1)32+3(xx2+1+1)(x2x2+1−1)(x+x2+1)x2+1+2(xx2+1+1)3(x+x2+1)2x+x2+1)x^{\sin{\left(x \right)}} \left(- \left(- \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3} + 3 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) + \log{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x} + \frac{3 \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin{\left(x \right)}}{x^{3}}\right) \log{\left(x + \sqrt{x^{2} + 1} \right)} + \frac{3 \left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right) \left(\left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} - \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}\right)}{x + \sqrt{x^{2} + 1}} - \frac{3 \left(\frac{\frac{x^{2}}{x^{2} + 1} - 1}{\sqrt{x^{2} + 1}} + \frac{\left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)^{2}}{x + \sqrt{x^{2} + 1}}\right) \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)}{x + \sqrt{x^{2} + 1}} + \frac{\frac{3 x \left(\frac{x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{3 \left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right) \left(\frac{x^{2}}{x^{2} + 1} - 1\right)}{\left(x + \sqrt{x^{2} + 1}\right) \sqrt{x^{2} + 1}} + \frac{2 \left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)^{3}}{\left(x + \sqrt{x^{2} + 1}\right)^{2}}}{x + \sqrt{x^{2} + 1}}\right)
Simplificar
Gráfico
Derivada de x^sinxln(x+(1+x^2)^1/2)
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