Answer
$\frac{dy}{dx}=\frac{3x^2}{2\sqrt{x^3+2}}$
Work Step by Step
Considering $u=x^3+2$, the function can be written as $y=\sqrt{u}$ or $y=u^{1/2}$.
We have the composite function $y=f(g(x))$ where $g(x)=x^3+2$ and $f(u)=u^{1/2}$.
Find $dy/du$:
$\frac{dy}{du}=\frac{d}{du}(u^{1/2})=\frac{1}{2}u^{-1/2}=\frac{1}{2\sqrt{u}}$
Find $du/dx$:
$\frac{du}{dx}=\frac{d}{dx}(x^3+2)=3x^2+0=3x^2$
Find $dy/dx$ using the chain rule:
$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}=\frac{1}{2\sqrt{u}}\cdot 3x^2=\frac{1}{2\sqrt{x^3+2}}\cdot 3x^2=\frac{3x^2}{2\sqrt{x^3+2}}$
Thus,
$\frac{dy}{dx}=\frac{3x^2}{2\sqrt{x^3+2}}$
