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