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(3^x)sin^-1x
Derivada de la función/ sin^-1x/ (3^x)sin^-1x

Derivada de (3^x)sin^-1x

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

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Solución

Ha introducido [src] 
   x  
  3   
------
sin(x)
3xsin⁡(x)\frac{3^{x}}{\sin{\left(x \right)}} 
3^x/sin(x)
Solución detallada

  1. Se aplica la regla de la derivada parcial:

    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)=3xf{\left(x \right)} = 3^{x} y g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. ddx3x=3xlog⁡(3)\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}

    Para calcular 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)}

    Ahora aplicamos la regla de la derivada de una divesión:

    3xlog⁡(3)sin⁡(x)−3xcos⁡(x)sin⁡2(x)\frac{3^{x} \log{\left(3 \right)} \sin{\left(x \right)} - 3^{x} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}

  2. Simplificamos:

    3x(log⁡(3)−1tan⁡(x))sin⁡(x)\frac{3^{x} \left(\log{\left(3 \right)} - \frac{1}{\tan{\left(x \right)}}\right)}{\sin{\left(x \right)}}

Respuesta:

3x(log⁡(3)−1tan⁡(x))sin⁡(x)\frac{3^{x} \left(\log{\left(3 \right)} - \frac{1}{\tan{\left(x \right)}}\right)}{\sin{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-5000000050000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
 x           x       
3 *log(3)   3 *cos(x)
--------- - ---------
  sin(x)        2    
             sin (x)
3xlog⁡(3)sin⁡(x)−3xcos⁡(x)sin⁡2(x)\frac{3^{x} \log{\left(3 \right)}}{\sin{\left(x \right)}} - \frac{3^{x} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}
Simplificar
Segunda derivada [src] 
   /                   2                     \
 x |       2      2*cos (x)   2*cos(x)*log(3)|
3 *|1 + log (3) + --------- - ---------------|
   |                  2            sin(x)    |
   \               sin (x)                   /
----------------------------------------------
                    sin(x)
3x(1+log⁡(3)2−2log⁡(3)cos⁡(x)sin⁡(x)+2cos⁡2(x)sin⁡2(x))sin⁡(x)\frac{3^{x} \left(1 + \log{\left(3 \right)}^{2} - \frac{2 \log{\left(3 \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\sin{\left(x \right)}}
Simplificar
Tercera derivada [src] 
   /                                     /         2   \                          \
   |                                     |    6*cos (x)|                          |
   |                                     |5 + ---------|*cos(x)                   |
   |            /         2   \          |        2    |               2          |
 x |   3        |    2*cos (x)|          \     sin (x) /          3*log (3)*cos(x)|
3 *|log (3) + 3*|1 + ---------|*log(3) - ---------------------- - ----------------|
   |            |        2    |                  sin(x)                sin(x)     |
   \            \     sin (x) /                                                   /
-----------------------------------------------------------------------------------
                                       sin(x)
3x(3(1+2cos⁡2(x)sin⁡2(x))log⁡(3)−(5+6cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x)+log⁡(3)3−3log⁡(3)2cos⁡(x)sin⁡(x))sin⁡(x)\frac{3^{x} \left(3 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(3 \right)} - \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(3 \right)}^{3} - \frac{3 \log{\left(3 \right)}^{2} \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{\sin{\left(x \right)}}
Simplificar
Gráfico
Derivada de (3^x)sin^-1x
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