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√xsin(x)*cos(x)
Derivada de la función/ √xsin/ cos(x)/ sin(x)/ √xsin(x)*cos(x)

Derivada de √xsin(x)*cos(x)

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

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Solución

Ha introducido [src] 
  ___              
\/ x *sin(x)*cos(x)
xsin⁡(x)cos⁡(x)\sqrt{x} \sin{\left(x \right)} \cos{\left(x \right)} 
(sqrt(x)*sin(x))*cos(x)
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)} = \sqrt{x} \sin{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    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)=xf{\left(x \right)} = \sqrt{x}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. La derivada del seno es igual al coseno:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      Como resultado de: xcos⁡(x)+sin⁡(x)2x\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}}

    g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. La derivada del coseno es igual a menos el seno:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    Como resultado de: −xsin⁡2(x)+(xcos⁡(x)+sin⁡(x)2x)cos⁡(x)- \sqrt{x} \sin^{2}{\left(x \right)} + \left(\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}}\right) \cos{\left(x \right)}

  2. Simplificamos:

    xcos⁡(2x)+sin⁡(2x)4x\frac{x \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{4}}{\sqrt{x}}

Respuesta:

xcos⁡(2x)+sin⁡(2x)4x\frac{x \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{4}}{\sqrt{x}}

Gráfica
02468-8-6-4-2-10105-5
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Primera derivada [src] 
/  ___           sin(x)\            ___    2   
|\/ x *cos(x) + -------|*cos(x) - \/ x *sin (x)
|                   ___|                       
\               2*\/ x /
−xsin⁡2(x)+(xcos⁡(x)+sin⁡(x)2x)cos⁡(x)- \sqrt{x} \sin^{2}{\left(x \right)} + \left(\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}}\right) \cos{\left(x \right)}
Simplificar
Segunda derivada [src] 
 /                                   /sin(x)   4*cos(x)       ___       \                             \
 |                                   |------ - -------- + 4*\/ x *sin(x)|*cos(x)                      |
 |                                   |  3/2       ___                   |                             |
 |/sin(x)       ___       \          \ x        \/ x                    /            ___              |
-||------ + 2*\/ x *cos(x)|*sin(x) + ------------------------------------------- + \/ x *cos(x)*sin(x)|
 ||  ___                  |                               4                                           |
 \\\/ x                   /                                                                           /
−(xsin⁡(x)cos⁡(x)+(2xcos⁡(x)+sin⁡(x)x)sin⁡(x)+(4xsin⁡(x)−4cos⁡(x)x+sin⁡(x)x32)cos⁡(x)4)- (\sqrt{x} \sin{\left(x \right)} \cos{\left(x \right)} + \left(2 \sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\sqrt{x}}\right) \sin{\left(x \right)} + \frac{\left(4 \sqrt{x} \sin{\left(x \right)} - \frac{4 \cos{\left(x \right)}}{\sqrt{x}} + \frac{\sin{\left(x \right)}}{x^{\frac{3}{2}}}\right) \cos{\left(x \right)}}{4})
Simplificar
Tercera derivada [src] 
                  /sin(x)       ___       \          /  3*sin(x)   6*cos(x)       ___          12*sin(x)\            /sin(x)   4*cos(x)       ___       \       
                3*|------ + 2*\/ x *cos(x)|*cos(x)   |- -------- + -------- + 8*\/ x *cos(x) + ---------|*cos(x)   3*|------ - -------- + 4*\/ x *sin(x)|*sin(x)
                  |  ___                  |          |     5/2        3/2                          ___  |            |  3/2       ___                   |       
  ___    2        \\/ x                   /          \    x          x                           \/ x   /            \ x        \/ x                    /       
\/ x *sin (x) - ---------------------------------- - ----------------------------------------------------------- + ---------------------------------------------
                                2                                                 8                                                      4
xsin⁡2(x)−3(2xcos⁡(x)+sin⁡(x)x)cos⁡(x)2+3(4xsin⁡(x)−4cos⁡(x)x+sin⁡(x)x32)sin⁡(x)4−(8xcos⁡(x)+12sin⁡(x)x+6cos⁡(x)x32−3sin⁡(x)x52)cos⁡(x)8\sqrt{x} \sin^{2}{\left(x \right)} - \frac{3 \left(2 \sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\sqrt{x}}\right) \cos{\left(x \right)}}{2} + \frac{3 \left(4 \sqrt{x} \sin{\left(x \right)} - \frac{4 \cos{\left(x \right)}}{\sqrt{x}} + \frac{\sin{\left(x \right)}}{x^{\frac{3}{2}}}\right) \sin{\left(x \right)}}{4} - \frac{\left(8 \sqrt{x} \cos{\left(x \right)} + \frac{12 \sin{\left(x \right)}}{\sqrt{x}} + \frac{6 \cos{\left(x \right)}}{x^{\frac{3}{2}}} - \frac{3 \sin{\left(x \right)}}{x^{\frac{5}{2}}}\right) \cos{\left(x \right)}}{8}
Simplificar
Gráfico
Derivada de √xsin(x)*cos(x)
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