https://r2.erweima.ai/mathbot/image/17256937199770xb5oi8t.jpg

The image contains a mathematical problem that involves partial derivatives.

### The Problem Statement
The equation given is:
\[
z^3 - 2yz - 8x = 0
\]

To show that:
\[
z \frac{\partial z}{\partial x} = \frac{\partial z}{\partial y},
\]

Then, a second part is given:
\[
(ii) \left[ z \frac{\partial^2 z}{\partial x \partial y} + \left( \frac{\partial z}{\partial x} \right)^2 \right] = \frac{\partial^2 z}{\partial y^2}
\]

### Steps to Solve

1. **Find Partial Derivatives of \(z\):**

   First, we need to differentiate the equation \(z^3 - 2yz - 8x = 0\) with respect to \(x\) and \(y\) to compute \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\).

2. **Differentiate with respect to \(x\):**

   \[
   \frac{\partial}{\partial x} (z^3 - 2yz - 8x) = 0
   \]

   Applying the chain rule:
   \[
   3z^2 \frac{\partial z}{\partial x} - 2y \frac{\partial z}{\partial x} - 8 = 0
   \]

   Rearranging to find \(\frac{\partial z}{\partial x}\):
   \[
   \frac{\partial z}{\partial x} (3z^2 - 2y) = 8
   \]

   \[
   \frac{\partial z}{\partial x} = \frac{8}{3z^2 - 2y}
   \]

3. **Differentiate with respect to \(y\):**

   \[
   \frac{\partial}{\partial y} (z^3 - 2yz - 8x) = 0
   \]

   Applying the chain rule:
   \[
   3z^2 \frac{\partial z}{\partial y} - 2z - 2y \frac{\partial z}{\partial y} = 0
   \]

   Rearranging to find \(\frac{\partial z}{\partial y}\):
   \[
   \frac{\partial z}{\partial y} (3z^2 - 2y) = 2z
   \]

   \[
   \frac{\partial z}{\partial y} = \frac{2z}{3z^2 - 2y}
   \]

4. **Verify the given expression:**

   Now we check if:
   \[
   z \frac{\partial z}{\partial x} = \frac{\partial z}{\partial y}
   \]

   Substituting the expressions found:
   \[
   z \left( \frac{8}{3z^2 - 2y} \right) = \frac{2z}{3z^2 - 2y}
   \]

   This simplifies to:
   \[
   \frac{8z}{3z^2 - 2y} = \frac{2z}{3z^2 - 2y}
   \]

   However, this appears inconsistent unless there is a mistake in the initial differentiation or assumptions. We might need to review the original partial derivatives carefully.

Would you like me to continue by re-evaluating the derivatives, or focus on the second part (ii) of the problem?

Solve for the partial derivatives from the equation: z^3 - 2yz - 8x = 0 and show that z ∂z/∂x = ∂z/∂y. Then solve the second part of the equation involving second-order partial derivatives.

Solve the partial derivatives for the equation z^3 - 2yz - 8x = 0 and prove the identity z ∂z/∂x = ∂z/∂y. This problem involves using the chain rule and differentiation techniques for multivariable calculus. It includes step-by-step solutions and second-order partial derivatives.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation