Question

derivada y=(1+raiz de x)3

Answer 1

$\displaystyle f(x)=(1+\sqrt{x})^3\implies f(x)=u^3\implies u(x)=1+\sqrt x \implies\\ (f\circ u)(x)=(1+\sqrt{x})^3\\\\\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}\ \left(regra~da~cadeia\right)\\\frac{d}{dx}(1+\sqrt{x})^3=\frac{d}{du}u^3\cdot\frac{d}{dx}1+\sqrt{x}\implies\frac{d}{dx}(1+\sqrt{x})^3=3u^2\cdot\frac{1}{2\sqrt{x}}\\\\\frac{d}{dx}(1+\sqrt{x})^3=\frac{3u^2}{2\sqrt{x}}\implies u=(1+\sqrt{x})\implies \frac{3u^2}{2\sqrt x}=\frac{3(1+\sqrt{x})^2}{2\sqrt{x}}\\\\\frac{df}{dx}=\frac{3(1+2\sqrt{x}+\sqrt{x}^2)}{2\sqrt{x}}=\frac{3(1+2\sqrt{x}+x)}{2\sqrt{x}}=\boxed{\frac{3+6\sqrt{x}+3x}{2\sqrt{x}}}$