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y=(3^x)/cosx
Derivada de la función/ y=(3^x)/ y=(3^x)/cosx

Derivada de y=(3^x)/cosx

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

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Solución

Ha introducido [src] 
   x  
  3   
------
cos(x)
3xcos⁡(x)\frac{3^{x}}{\cos{\left(x \right)}} 
3^x/cos(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)=cos⁡(x)g{\left(x \right)} = \cos{\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 coseno es igual a menos el seno:

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

    Ahora aplicamos la regla de la derivada de una divesión:

    3xsin⁡(x)+3xlog⁡(3)cos⁡(x)cos⁡2(x)\frac{3^{x} \sin{\left(x \right)} + 3^{x} \log{\left(3 \right)} \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Simplificamos:

    3x(tan⁡(x)+log⁡(3))cos⁡(x)\frac{3^{x} \left(\tan{\left(x \right)} + \log{\left(3 \right)}\right)}{\cos{\left(x \right)}}

Respuesta:

3x(tan⁡(x)+log⁡(3))cos⁡(x)\frac{3^{x} \left(\tan{\left(x \right)} + \log{\left(3 \right)}\right)}{\cos{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-10000001000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
 x           x       
3 *log(3)   3 *sin(x)
--------- + ---------
  cos(x)        2    
             cos (x)
3xsin⁡(x)cos⁡2(x)+3xlog⁡(3)cos⁡(x)\frac{3^{x} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3^{x} \log{\left(3 \right)}}{\cos{\left(x \right)}}
Simplificar
Segunda derivada [src] 
   /                   2                     \
 x |       2      2*sin (x)   2*log(3)*sin(x)|
3 *|1 + log (3) + --------- + ---------------|
   |                  2            cos(x)    |
   \               cos (x)                   /
----------------------------------------------
                    cos(x)
3x(2sin⁡2(x)cos⁡2(x)+2log⁡(3)sin⁡(x)cos⁡(x)+1+log⁡(3)2)cos⁡(x)\frac{3^{x} \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 \log{\left(3 \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \log{\left(3 \right)}^{2}\right)}{\cos{\left(x \right)}}
Simplificar
Tercera derivada [src] 
   /                                     /         2   \                          \
   |                                     |    6*sin (x)|                          |
   |                                     |5 + ---------|*sin(x)                   |
   |            /         2   \          |        2    |               2          |
 x |   3        |    2*sin (x)|          \     cos (x) /          3*log (3)*sin(x)|
3 *|log (3) + 3*|1 + ---------|*log(3) + ---------------------- + ----------------|
   |            |        2    |                  cos(x)                cos(x)     |
   \            \     cos (x) /                                                   /
-----------------------------------------------------------------------------------
                                       cos(x)
3x(3(2sin⁡2(x)cos⁡2(x)+1)log⁡(3)+(6sin⁡2(x)cos⁡2(x)+5)sin⁡(x)cos⁡(x)+3log⁡(3)2sin⁡(x)cos⁡(x)+log⁡(3)3)cos⁡(x)\frac{3^{x} \left(3 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \log{\left(3 \right)} + \frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \log{\left(3 \right)}^{2} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(3 \right)}^{3}\right)}{\cos{\left(x \right)}}
Simplificar
Gráfico
Derivada de y=(3^x)/cosx
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