To solve this problem, we need to break it down into several steps:

1. Expression for $u$:

The given expression for $u$ is: $u = \sin^{-1} \left( \frac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}} \right)$ This means $u$ is the inverse sine (arcsine) of the given ratio.

2. Applying Euler's Theorem:

Euler's theorem for sine in terms of complex exponentials is: $e^{i\theta} = \cos\theta + i\sin\theta$ Thus: $\sin u = \frac{e^{iu} - e^{-iu}}{2i}$ However, here we are not directly concerned with expressing $\sin u$ in terms of exponentials, since the question primarily revolves around the derivatives. But Euler’s identity can remind us how trigonometric functions are related to complex exponentials.

3. Implicit Differentiation:

We need to compute second-order partial derivatives of $u$ with respect to $x$ and $y$.

Let’s define: $z = \frac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}}$ Thus: $u = \sin^{-1}(z)$ To find the partial derivatives $u_x$, $u_y$, and second-order partials $u_{xx}$, $u_{xy}$, and $u_{yy}$, we will differentiate $u$ with respect to $x$ and $y$.

4. First-Order Partial Derivatives:

The chain rule for $u = \sin^{-1}(z)$ gives: $u_x = \frac{1}{\sqrt{1 - z^2}} \cdot z_x$ $u_y = \frac{1}{\sqrt{1 - z^2}} \cdot z_y$ We now need to calculate $z_x$ and $z_y$.

5. Second-Order Partial Derivatives:

For the second-order partial derivatives $u_{xx}$, $u_{xy}$, and $u_{yy}$, we apply the product rule and chain rule carefully. For example, for $u_{xx}$: $u_{xx} = \frac{\partial}{\partial x} \left( \frac{1}{\sqrt{1 - z^2}} \cdot z_x \right)$ This involves taking the derivative of both terms (using the product and chain rules).

6. Substituting into the Original Equation:

The goal is to show that: $x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy} = \frac{\tan u}{12} + \frac{\tan^2 u}{12}$ After calculating the second-order partial derivatives and substituting them into the left-hand side, you should get the required expression.

Summary of Approach:

• Step 1: Write $u$ in terms of arcsine and the function $z$.
• Step 2: Use chain rule to find $u_x$ and $u_y$.
• Step 3: Use chain rule and product rule to find second-order derivatives $u_{xx}$, $u_{xy}$, and $u_{yy}$.
• Step 4: Substitute the second-order partial derivatives into the given equation and simplify to get the desired result.

Would you like a detailed step-by-step calculation for each derivative?

5 Questions:

1. Would you like to see the full steps for finding $u_x$ and $u_y$?
2. Are you interested in the explicit forms of $z_x$ and $z_y$?
3. Would you like help simplifying the second-order partial derivatives?
4. Would you like an explanation of how Euler's Theorem is applied here?
5. Are there other identities you'd like to explore for the arcsine function?

Tip:

When differentiating implicit functions like $u = \sin^{-1}(z)$, always remember to apply the chain rule for both first- and second-order derivatives to ensure accurate results.