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y=(tgx)/(4^x)
Derivada de la función/ (tgx)/ y=(tgx)/(4^x)

Derivada de y=(tgx)/(4^x)

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

Ha introducido [src] 
tan(x)
------
   x  
  4
tan⁡(x)4x\frac{\tan{\left(x \right)}}{4^{x}} 
tan(x)/4^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)=tan⁡(x)f{\left(x \right)} = \tan{\left(x \right)} y g(x)=4xg{\left(x \right)} = 4^{x}.

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

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. ddx4x=4xlog⁡(4)\frac{d}{d x} 4^{x} = 4^{x} \log{\left(4 \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    4−2x(4x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−4xlog⁡(4)tan⁡(x))4^{- 2 x} \left(\frac{4^{x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 4^{x} \log{\left(4 \right)} \tan{\left(x \right)}\right)

  2. Simplificamos:

    4−x(−2log⁡(2)tan⁡(x)+1cos⁡2(x))4^{- x} \left(- 2 \log{\left(2 \right)} \tan{\left(x \right)} + \frac{1}{\cos^{2}{\left(x \right)}}\right)

Respuesta:

4−x(−2log⁡(2)tan⁡(x)+1cos⁡2(x))4^{- x} \left(- 2 \log{\left(2 \right)} \tan{\left(x \right)} + \frac{1}{\cos^{2}{\left(x \right)}}\right)

Gráfica
02468-8-6-4-2-1010-1000000010000000
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Trazar el gráfico f'(x)
Primera derivada [src] 
 -x /       2   \    -x              
4  *\1 + tan (x)/ - 4  *log(4)*tan(x)
4−x(tan⁡2(x)+1)−4−xlog⁡(4)tan⁡(x)4^{- x} \left(\tan^{2}{\left(x \right)} + 1\right) - 4^{- x} \log{\left(4 \right)} \tan{\left(x \right)}
Simplificar
Segunda derivada [src] 
 -x /   2               /       2   \            /       2   \       \
4  *\log (4)*tan(x) - 2*\1 + tan (x)/*log(4) + 2*\1 + tan (x)/*tan(x)/
4−x(2(tan⁡2(x)+1)tan⁡(x)−2(tan⁡2(x)+1)log⁡(4)+log⁡(4)2tan⁡(x))4^{- 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(4 \right)} + \log{\left(4 \right)}^{2} \tan{\left(x \right)}\right)
Simplificar
Tercera derivada [src] 
 -x /     3               /       2   \ /         2   \        2    /       2   \     /       2   \              \
4  *\- log (4)*tan(x) + 2*\1 + tan (x)/*\1 + 3*tan (x)/ + 3*log (4)*\1 + tan (x)/ - 6*\1 + tan (x)/*log(4)*tan(x)/
4−x(2(tan⁡2(x)+1)(3tan⁡2(x)+1)−6(tan⁡2(x)+1)log⁡(4)tan⁡(x)+3(tan⁡2(x)+1)log⁡(4)2−log⁡(4)3tan⁡(x))4^{- 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(4 \right)} \tan{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(4 \right)}^{2} - \log{\left(4 \right)}^{3} \tan{\left(x \right)}\right)
Simplificar
Gráfico
Derivada de y=(tgx)/(4^x)
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