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