If u=(x^2+y^2/x+y),show that (du/dx-du/dy)^2 = 4(1-du/dx-du/dy)

Given the expression for \( u \):

\[
u = \frac{x^2 + \frac{y^2}{x}}{y}
\]

We need to show that:

\[
\left(\frac{du}{dx} - \frac{du}{dy}\right)^2 = 4 \left(1 - \frac{du}{dx} - \frac{du}{dy}\right)
\]

### Step 1: Compute \(\frac{du}{dx}\)
We differentiate \( u \) with respect to \( x \), applying the quotient rule and chain rule as needed.

Since the expression for \( u \) contains both \( x \) and \( y \), we proceed by first finding partial derivatives.

### Step 2: Compute \(\frac{du}{dy}\)


If u = (x^2 + y^2) / (x + y), show that (du/dx - du/dy)^2 = 4(1 - du/dx - du/dy)

Explore a calculus problem where u = (x^2 + y^2) / (x + y) and learn how to prove that (du/dx - du/dy)^2 = 4(1 - du/dx - du/dy). This problem utilizes partial derivatives, the quotient rule, and the chain rule. Suitable for college-level students.

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