Hallar $$$\frac{d^{5}}{dx^{5}} \left(\sin{\left(x \right)}\right)$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d^{5}}{dx^{5}} \left(\sin{\left(x \right)}\right)$$$.
Solución
Calcule la primera derivada $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right)$$$
La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = {\color{red}\left(\cos{\left(x \right)}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$.
A continuación, $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$$
La derivada del coseno es $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = {\color{red}\left(- \sin{\left(x \right)}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$.
A continuación, $$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \sin{\left(x \right)}\right)$$$
Aplica la regla del factor constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = -1$$$ y $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \sin{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = - {\color{red}\left(\cos{\left(x \right)}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(- \sin{\left(x \right)}\right) = - \cos{\left(x \right)}$$$.
A continuación, $$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \cos{\left(x \right)}\right)$$$
Aplica la regla del factor constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = -1$$$ y $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \cos{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$La derivada del coseno es $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = - {\color{red}\left(- \sin{\left(x \right)}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(- \cos{\left(x \right)}\right) = \sin{\left(x \right)}$$$.
A continuación, $$$\frac{d^{5}}{dx^{5}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$$
La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = {\color{red}\left(\cos{\left(x \right)}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$.
Por lo tanto, $$$\frac{d^{5}}{dx^{5}} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$.
Respuesta
$$$\frac{d^{5}}{dx^{5}} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$A