Derivado de $$$\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de diferenciación implícita con pasos
Tu aportación
Encuentra $$$\frac{d}{dx} \left(\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}\right)$$$.
Solución
La función $$$\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \sqrt[9]{u}$$$ y $$$g{\left(x \right)} = x^{2} + 3 + \frac{2}{x^{2}}$$$.
Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sqrt[9]{u}\right) \frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ con $$$n = \frac{1}{9}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sqrt[9]{u}\right)\right)} \frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right) = {\color{red}\left(\frac{1}{9 u^{\frac{8}{9}}}\right)} \frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)$$Vuelva a la variable anterior:
$$\frac{\frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)}{9 {\color{red}\left(u\right)}^{\frac{8}{9}}} = \frac{\frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)}{9 {\color{red}\left(x^{2} + 3 + \frac{2}{x^{2}}\right)}^{\frac{8}{9}}}$$La derivada de una suma/diferencia es la suma/diferencia de derivadas:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right) + \frac{d}{dx} \left(3\right) + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(3\right) + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{{\color{red}\left(2 x\right)} + \frac{d}{dx} \left(3\right) + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$La derivada de una constante es $$$0$$$:
$$\frac{2 x + {\color{red}\left(\frac{d}{dx} \left(3\right)\right)} + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{2 x + {\color{red}\left(0\right)} + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 2$$$ y $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:
$$\frac{2 x + {\color{red}\left(\frac{d}{dx} \left(\frac{2}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{2 x + {\color{red}\left(2 \frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = -2$$$:
$$\frac{2 x + 2 {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{2 x + 2 {\color{red}\left(- \frac{2}{x^{3}}\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$Por lo tanto, $$$\frac{d}{dx} \left(\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}\right) = \frac{2 x - \frac{4}{x^{3}}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$$.
Respuesta
$$$\frac{d}{dx} \left(\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}\right) = \frac{2 x - \frac{4}{x^{3}}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$$A