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Derivada de $$$\sec^{3}{\left(u \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right)$$$.
Solución
La función $$$\sec^{3}{\left(u \right)}$$$ es la composición $$$f{\left(g{\left(u \right)} \right)}$$$ de dos funciones $$$f{\left(v \right)} = v^{3}$$$ y $$$g{\left(u \right)} = \sec{\left(u \right)}$$$.
Aplica la regla de la cadena $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$:
$${\color{red}\left(\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(v^{3}\right) \frac{d}{du} \left(\sec{\left(u \right)}\right)\right)}$$Aplica la regla de la potencia $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$ con $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dv} \left(v^{3}\right)\right)} \frac{d}{du} \left(\sec{\left(u \right)}\right) = {\color{red}\left(3 v^{2}\right)} \frac{d}{du} \left(\sec{\left(u \right)}\right)$$Volver a la variable original:
$$3 {\color{red}\left(v\right)}^{2} \frac{d}{du} \left(\sec{\left(u \right)}\right) = 3 {\color{red}\left(\sec{\left(u \right)}\right)}^{2} \frac{d}{du} \left(\sec{\left(u \right)}\right)$$La derivada de la secante es $$$\frac{d}{du} \left(\sec{\left(u \right)}\right) = \tan{\left(u \right)} \sec{\left(u \right)}$$$:
$$3 \sec^{2}{\left(u \right)} {\color{red}\left(\frac{d}{du} \left(\sec{\left(u \right)}\right)\right)} = 3 \sec^{2}{\left(u \right)} {\color{red}\left(\tan{\left(u \right)} \sec{\left(u \right)}\right)}$$Por lo tanto, $$$\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right) = 3 \tan{\left(u \right)} \sec^{3}{\left(u \right)}$$$.
Respuesta
$$$\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right) = 3 \tan{\left(u \right)} \sec^{3}{\left(u \right)}$$$A