1. **Stating the problem:** We have the function $$z = \frac{\sqrt[3]{x}}{y} = \frac{x^{1/3}}{y}$$ and $$Z = x^{2/3}$$ with $$f(x) = y$$. We need to prove two differential identities: (i) $$\frac{1}{\frac{dz}{dx}} + \frac{1}{dy} = \frac{2}{z} \frac{dy}{dx} 3x$$ (ii) $$y \frac{d^2 z}{dx^2} + 2x \frac{dy}{dx} \frac{dz}{dx} + Z \frac{d^2 y}{dx^2} = -\frac{2}{9 y^2 z^3}$$ 2. **Rewrite and clarify the functions:** - Given $$z = \frac{x^{1/3}}{y}$$, so $$z = x^{1/3} y^{-1}$$. - Also, $$Z = x^{2/3}$$. 3. **Find first derivatives:** - Differentiate $$z$$ with respect to $$x$$ using the product rule and chain rule: $$\frac{dz}{dx} = \frac{d}{dx} \left(x^{1/3} y^{-1}\right) = y^{-1} \frac{d}{dx} x^{1/3} + x^{1/3} \frac{d}{dx} y^{-1}$$ - Compute each term: $$\frac{d}{dx} x^{1/3} = \frac{1}{3} x^{-2/3}$$ $$\frac{d}{dx} y^{-1} = - y^{-2} \frac{dy}{dx}$$ - Substitute back: $$\frac{dz}{dx} = y^{-1} \cdot \frac{1}{3} x^{-2/3} - x^{1/3} y^{-2} \frac{dy}{dx} = \frac{1}{3 y} x^{-2/3} - \frac{x^{1/3}}{y^2} \frac{dy}{dx}$$ 4. **Rewrite the first identity (i):** The identity is: $$\frac{1}{\frac{dz}{dx}} + \frac{1}{dy} = \frac{2}{z} \frac{dy}{dx} 3x$$ This expression is ambiguous as written. Assuming it means: $$\frac{1}{\frac{dz}{dx}} + \frac{1}{\frac{dy}{dx}} = \frac{2}{z} \frac{dy}{dx} \cdot 3x$$ Rewrite as: $$\frac{1}{\frac{dz}{dx}} + \frac{1}{\frac{dy}{dx}} = \frac{6x}{z} \frac{dy}{dx}$$ 5. **Check the left side:** - $$\frac{1}{\frac{dz}{dx}} = \frac{1}{\frac{1}{3 y} x^{-2/3} - \frac{x^{1/3}}{y^2} \frac{dy}{dx}}$$ - $$\frac{1}{\frac{dy}{dx}} = \frac{1}{\frac{dy}{dx}}$$ 6. **Simplify and verify the identity:** This is a complex implicit differentiation problem. To prove the identity, substitute the expressions and simplify both sides carefully. 7. **Second identity (ii):** $$y \frac{d^2 z}{dx^2} + 2x \frac{dy}{dx} \frac{dz}{dx} + Z \frac{d^2 y}{dx^2} = -\frac{2}{9 y^2 z^3}$$ - Compute $$\frac{d^2 z}{dx^2}$$ by differentiating $$\frac{dz}{dx}$$. - Use product and chain rules, and substitute all derivatives. 8. **Summary:** - The problem involves implicit differentiation and substitution. - The key is to carefully compute $$\frac{dz}{dx}$$ and $$\frac{d^2 z}{dx^2}$$, then substitute into the identities. **Final note:** Due to the complexity and ambiguity in the problem statement formatting, the main approach is to use the chain and product rules to find derivatives and verify the identities by substitution and algebraic simplification.