Derivada de $$$\operatorname{atan}{\left(\frac{1}{x} \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(\operatorname{atan}{\left(\frac{1}{x} \right)}\right)$$$.
Solución
La función $$$\operatorname{atan}{\left(\frac{1}{x} \right)}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$ y $$$g{\left(x \right)} = \frac{1}{x}$$$.
Aplica la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\operatorname{atan}{\left(\frac{1}{x} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) \frac{d}{dx} \left(\frac{1}{x}\right)\right)}$$La derivada de la arctangente es $$$\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) = \frac{1}{u^{2} + 1}$$$:
$${\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{1}{x}\right) = {\color{red}\left(\frac{1}{u^{2} + 1}\right)} \frac{d}{dx} \left(\frac{1}{x}\right)$$Volver a la variable original:
$$\frac{\frac{d}{dx} \left(\frac{1}{x}\right)}{{\color{red}\left(u\right)}^{2} + 1} = \frac{\frac{d}{dx} \left(\frac{1}{x}\right)}{{\color{red}\left(\frac{1}{x}\right)}^{2} + 1}$$Aplica la regla de la potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = -1$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)}}{1 + \frac{1}{x^{2}}} = \frac{{\color{red}\left(- \frac{1}{x^{2}}\right)}}{1 + \frac{1}{x^{2}}}$$Simplificar:
$$- \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)} = - \frac{1}{x^{2} + 1}$$Por lo tanto, $$$\frac{d}{dx} \left(\operatorname{atan}{\left(\frac{1}{x} \right)}\right) = - \frac{1}{x^{2} + 1}$$$.
Respuesta
$$$\frac{d}{dx} \left(\operatorname{atan}{\left(\frac{1}{x} \right)}\right) = - \frac{1}{x^{2} + 1}$$$A