Derivada de $$$x^{3} + 5 x^{2} + 7 x + 4$$$
Calculadoras relacionadas: Calculadora de diferenciação logarítmica, Calculadora de Diferenciação Implícita com Passos
Sua entrada
Encontre $$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right)$$$.
Solução
A derivada de uma soma/diferença é a soma/diferença das 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 a regra múltipla constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ com $$$c = 5$$$ e $$$f{\left(x \right)} = x^{2}$$$:
$${\color{red}\left(\frac{d}{dx} \left(5 x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right) = {\color{red}\left(5 \frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right)$$Aplique a regra de poder $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 2$$$:
$$5 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right) = 5 {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(x^{3}\right)$$Aplique a regra de poder $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 3$$$:
$$10 x + {\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) = 10 x + {\color{red}\left(3 x^{2}\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right)$$Aplique a regra múltipla constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ com $$$c = 7$$$ e $$$f{\left(x \right)} = x$$$:
$$3 x^{2} + 10 x + {\color{red}\left(\frac{d}{dx} \left(7 x\right)\right)} + \frac{d}{dx} \left(4\right) = 3 x^{2} + 10 x + {\color{red}\left(7 \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(4\right)$$Aplique a regra de potência $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 1$$$, ou seja, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$3 x^{2} + 10 x + 7 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(4\right) = 3 x^{2} + 10 x + 7 {\color{red}\left(1\right)} + \frac{d}{dx} \left(4\right)$$A derivada de uma constante é $$$0$$$:
$$3 x^{2} + 10 x + {\color{red}\left(\frac{d}{dx} \left(4\right)\right)} + 7 = 3 x^{2} + 10 x + {\color{red}\left(0\right)} + 7$$Simplificar:
$$3 x^{2} + 10 x + 7 = \left(x + 1\right) \left(3 x + 7\right)$$Assim, $$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)$$$.
Responder
$$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)$$$A