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Derivado de $$$\tan{\left(x^{2} \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de diferenciación implícita con pasos
Tu aportación
Encuentra $$$\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right)$$$.
Solución
La función $$$\tan{\left(x^{2} \right)}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \tan{\left(u \right)}$$$ y $$$g{\left(x \right)} = x^{2}$$$.
Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\tan{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$La derivada de la tangente es $$$\frac{d}{du} \left(\tan{\left(u \right)}\right) = \sec^{2}{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\tan{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\sec^{2}{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right)$$Vuelva a la variable anterior:
$$\sec^{2}{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) = \sec^{2}{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right)$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:
$$\sec^{2}{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = \sec^{2}{\left(x^{2} \right)} {\color{red}\left(2 x\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right) = 2 x \sec^{2}{\left(x^{2} \right)}$$$.
Respuesta
$$$\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right) = 2 x \sec^{2}{\left(x^{2} \right)}$$$A