To verify that $\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}$ for the function $z = \sin^{-1}(xy)$, let's compute both mixed partial derivatives and see if they are indeed equal.

Step 1: Compute the First Partial Derivatives

Given: $z = \sin^{-1}(xy)$

1. Compute $\frac{\partial z}{\partial x}$: $\frac{\partial z}{\partial x} = \frac{1}{\sqrt{1 - (xy)^2}} \cdot y = \frac{y}{\sqrt{1 - x^2 y^2}}$

2. Compute $\frac{\partial z}{\partial y}$: $\frac{\partial z}{\partial y} = \frac{1}{\sqrt{1 - (xy)^2}} \cdot x = \frac{x}{\sqrt{1 - x^2 y^2}}$

Step 2: Compute the Second Mixed Partial Derivatives

Now, we compute $\frac{\partial^2 z}{\partial y \partial x}$ and $\frac{\partial^2 z}{\partial x \partial y}$ to verify if they are equal.

1. Compute $\frac{\partial^2 z}{\partial y \partial x}$: $\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial}{\partial y} \left( \frac{y}{\sqrt{1 - x^2 y^2}} \right)$ Using the product rule and the chain rule, we get: $= \frac{1}{\sqrt{1 - x^2 y^2}} + y \cdot \frac{x^2 y}{\left( 1 - x^2 y^2 \right)^{3/2}}$ Simplifying, we get: $\frac{\partial^2 z}{\partial y \partial x} = \frac{1}{\sqrt{1 - x^2 y^2}} + \frac{x^2 y^2}{\left( 1 - x^2 y^2 \right)^{3/2}}$

2. Compute $\frac{\partial^2 z}{\partial x \partial y}$: $\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial x} \left( \frac{x}{\sqrt{1 - x^2 y^2}} \right)$ Similarly, using the product rule and the chain rule, we get: $= \frac{1}{\sqrt{1 - x^2 y^2}} + x \cdot \frac{x y^2}{\left( 1 - x^2 y^2 \right)^{3/2}}$ Simplifying, we get: $\frac{\partial^2 z}{\partial x \partial y} = \frac{1}{\sqrt{1 - x^2 y^2}} + \frac{x^2 y^2}{\left( 1 - x^2 y^2 \right)^{3/2}}$

Conclusion

Since: $\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y} = \frac{1}{\sqrt{1 - x^2 y^2}} + \frac{x^2 y^2}{\left( 1 - x^2 y^2 \right)^{3/2}}$ we have verified that the mixed partial derivatives are indeed equal.

Would you like further details on any of these steps?

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1. What are the conditions under which mixed partial derivatives are equal?
2. How can the product and chain rules be applied to complex derivatives?
3. Can you verify this result using another trigonometric function for $z$?
4. What happens if $z$ is a function of three variables instead of two?
5. How do we interpret these derivatives geometrically?

Tip: Always check continuity and differentiability when verifying the equality of mixed partial derivatives.