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

Derivada de y=(3^x)×tanx

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

Ha introducido [src] 
 x       
3 *tan(x)
3xtan⁡(x)3^{x} \tan{\left(x \right)} 
3^x*tan(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)=3xf{\left(x \right)} = 3^{x}; calculamos 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)}

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

    1. Reescribimos las funciones para diferenciar:

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

    2. 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)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} 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. La derivada del seno es igual al coseno:

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

      sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    Como resultado de: 3x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+3xlog⁡(3)tan⁡(x)\frac{3^{x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 3^{x} \log{\left(3 \right)} \tan{\left(x \right)}

  2. Simplificamos:

    3x(log⁡(3)sin⁡(2x)2+1)cos⁡2(x)\frac{3^{x} \left(\frac{\log{\left(3 \right)} \sin{\left(2 x \right)}}{2} + 1\right)}{\cos^{2}{\left(x \right)}}

Respuesta:

3x(log⁡(3)sin⁡(2x)2+1)cos⁡2(x)\frac{3^{x} \left(\frac{\log{\left(3 \right)} \sin{\left(2 x \right)}}{2} + 1\right)}{\cos^{2}{\left(x \right)}}

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