Derivada de y=4^x\cosx

Solución

Ha introducido [src]

   x  
  4   
------
cos(x)

$$\frac{4^{x}}{\cos{\left(x \right)}}$$

4^x/cos(x)

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 4^{x}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. $\frac{d}{d x} 4^{x} = 4^{x} \log{\left(4 \right)}$
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:

$\frac{4^{x} \sin{\left(x \right)} + 4^{x} \log{\left(4 \right)} \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Simplificamos:

$\frac{4^{x} \left(\log{\left(4^{\cos{\left(x \right)}} \right)} + \sin{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$

Respuesta:

$\frac{4^{x} \left(\log{\left(4^{\cos{\left(x \right)}} \right)} + \sin{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

 x           x       
4 *log(4)   4 *sin(x)
--------- + ---------
  cos(x)        2    
             cos (x)

$$\frac{4^{x} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{4^{x} \log{\left(4 \right)}}{\cos{\left(x \right)}}$$

Simplificar

Segunda derivada [src]

   /                   2                     \
 x |       2      2*sin (x)   2*log(4)*sin(x)|
4 *|1 + log (4) + --------- + ---------------|
   |                  2            cos(x)    |
   \               cos (x)                   /
----------------------------------------------
                    cos(x)

$$\frac{4^{x} \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 \log{\left(4 \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \log{\left(4 \right)}^{2}\right)}{\cos{\left(x \right)}}$$

Simplificar

Tercera derivada [src]

   /                                     /         2   \                          \
   |                                     |    6*sin (x)|                          |
   |                                     |5 + ---------|*sin(x)                   |
   |            /         2   \          |        2    |               2          |
 x |   3        |    2*sin (x)|          \     cos (x) /          3*log (4)*sin(x)|
4 *|log (4) + 3*|1 + ---------|*log(4) + ---------------------- + ----------------|
   |            |        2    |                  cos(x)                cos(x)     |
   \            \     cos (x) /                                                   /
-----------------------------------------------------------------------------------
                                       cos(x)

$$\frac{4^{x} \left(3 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \log{\left(4 \right)} + \frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \log{\left(4 \right)}^{2} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(4 \right)}^{3}\right)}{\cos{\left(x \right)}}$$

Simplificar

Gráfico

$Derivada de y=4^x\cosx$

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Derivada de y=4^x\cosx

Gráfico:

Definida a trozos:

Solución