Derivada de y=(3^x)×tanx

Solución

Ha introducido [src]

 x       
3 *tan(x)

$$3^{x} \tan{\left(x \right)}$$

3^x*tan(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = 3^{x}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. $\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Reescribimos las funciones para diferenciar:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. La derivada del coseno es igual a menos el seno:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Como resultado de: $\frac{3^{x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 3^{x} \log{\left(3 \right)} \tan{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

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Primera derivada [src]

 x /       2   \    x              
3 *\1 + tan (x)/ + 3 *log(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)/

$$3^{x} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} + \log{\left(3 \right)}^{2} \tan{\left(x \right)}\right)$$

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)/

$$3^{x} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} \tan{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)}^{2} + \log{\left(3 \right)}^{3} \tan{\left(x \right)}\right)$$

Simplificar

Gráfico

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Derivada de y=(3^x)×tanx

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