Derivado de $$$x^{3} + 5 x^{2} + 7 x + 4$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de diferenciación implícita con pasos
Tu aportación
Encuentra $$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right)$$$.
Solución
La derivada de una suma/diferencia es la suma/diferencia de derivadas:
$${\color{red}\left(\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) + \frac{d}{dx} \left(5 x^{2}\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(4\right)\right)}$$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 $$$c = 5$$$ y $$$f{\left(x \right)} = x^{2}$$$:
$${\color{red}\left(\frac{d}{dx} \left(5 x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right) = {\color{red}\left(5 \frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right)$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:
$$5 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right) = 5 {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right)$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 3$$$:
$$10 x + {\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) = 10 x + {\color{red}\left(3 x^{2}\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right)$$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 $$$c = 7$$$ y $$$f{\left(x \right)} = x$$$:
$$3 x^{2} + 10 x + {\color{red}\left(\frac{d}{dx} \left(7 x\right)\right)} + \frac{d}{dx} \left(4\right) = 3 x^{2} + 10 x + {\color{red}\left(7 \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(4\right)$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$3 x^{2} + 10 x + 7 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(4\right) = 3 x^{2} + 10 x + 7 {\color{red}\left(1\right)} + \frac{d}{dx} \left(4\right)$$La derivada de una constante es $$$0$$$:
$$3 x^{2} + 10 x + {\color{red}\left(\frac{d}{dx} \left(4\right)\right)} + 7 = 3 x^{2} + 10 x + {\color{red}\left(0\right)} + 7$$Simplificar:
$$3 x^{2} + 10 x + 7 = \left(x + 1\right) \left(3 x + 7\right)$$Por lo tanto, $$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)$$$.
Respuesta
$$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)$$$A