Derivada de y=arcctg10^x*cosx

Solución

Ha introducido [src]

    x           
acot (10)*cos(x)

$$\cos{\left(x \right)} \operatorname{acot}^{x}{\left(10 \right)}$$

acot(10)^x*cos(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \operatorname{acot}^{x}{\left(10 \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. $\frac{d}{d x} \operatorname{acot}^{x}{\left(10 \right)} = \log{\left(\operatorname{acot}{\left(10 \right)} \right)} \operatorname{acot}^{x}{\left(10 \right)}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. La derivada del coseno es igual a menos el seno:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
Como resultado de: $- \sin{\left(x \right)} \operatorname{acot}^{x}{\left(10 \right)} + \log{\left(\operatorname{acot}{\left(10 \right)} \right)} \cos{\left(x \right)} \operatorname{acot}^{x}{\left(10 \right)}$
Simplificamos:

$\left(- \sin{\left(x \right)} + \log{\left(\operatorname{acot}{\left(10 \right)} \right)} \cos{\left(x \right)}\right) \operatorname{acot}^{x}{\left(10 \right)}$

Respuesta:

$\left(- \sin{\left(x \right)} + \log{\left(\operatorname{acot}{\left(10 \right)} \right)} \cos{\left(x \right)}\right) \operatorname{acot}^{x}{\left(10 \right)}$

Primera derivada [src]

      x                  x                         
- acot (10)*sin(x) + acot (10)*cos(x)*log(acot(10))

$$- \sin{\left(x \right)} \operatorname{acot}^{x}{\left(10 \right)} + \log{\left(\operatorname{acot}{\left(10 \right)} \right)} \cos{\left(x \right)} \operatorname{acot}^{x}{\left(10 \right)}$$

Simplificar

Segunda derivada [src]

    x     /             2                                          \
acot (10)*\-cos(x) + log (acot(10))*cos(x) - 2*log(acot(10))*sin(x)/

$$\left(- 2 \log{\left(\operatorname{acot}{\left(10 \right)} \right)} \sin{\left(x \right)} - \cos{\left(x \right)} + \log{\left(\operatorname{acot}{\left(10 \right)} \right)}^{2} \cos{\left(x \right)}\right) \operatorname{acot}^{x}{\left(10 \right)}$$

Simplificar

Tercera derivada [src]

    x     /   3                         2                                                   \
acot (10)*\log (acot(10))*cos(x) - 3*log (acot(10))*sin(x) - 3*cos(x)*log(acot(10)) + sin(x)/

$$\left(- 3 \log{\left(\operatorname{acot}{\left(10 \right)} \right)}^{2} \sin{\left(x \right)} + \sin{\left(x \right)} + \log{\left(\operatorname{acot}{\left(10 \right)} \right)}^{3} \cos{\left(x \right)} - 3 \log{\left(\operatorname{acot}{\left(10 \right)} \right)} \cos{\left(x \right)}\right) \operatorname{acot}^{x}{\left(10 \right)}$$

Simplificar

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Derivada de y=arcctg10^x*cosx

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