91Ó°ÊÓ

Partial differentiation is a technique used to differentiate a function with respect to one variable while keeping the other variables constant.
For example, in the function \( f(x, y) = x \, \text{ln}(x - y) \) , if we take the partial derivative with respect to \( x \), we maintain \( y \) as a constant.
This method allows us to see how the function changes as we tweak one variable at a time.
When finding partial derivatives like \( f_x \) and \( f_y \), we focus on just one variable's influence on the function's output.
It's crucial to approach this step with care to ensure accuracy.

The product rule is a fundamental technique to differentiate functions that are products of two or more functions.
If we have functions \( u \) and \( v \), their product \( uv \) is differentiated as follows: \[ \frac{d}{dx} [uv] = u'v + uv' \]
Don't forget, each time we multiply the derivatives by their respective original functions.
For our exercise, with \( u = x \) and \( v = \text{ln}(x - y) \), we get:\[ f_x = 1 \cdot \text{ln}(x-y) + x \cdot \frac{1}{x-y} \]
This gives us \( f_x = \text{ln}(x-y) + \frac{x}{x-y} \).

The chain rule helps differentiate composite functions by breaking down their layers.
When a function is nested within another, we apply the chain rule to differentiate.
In the given function, to find the partial derivative \( f_y \), we use the chain rule:
If we have a function \( g(h(y)) \), it's differentiated as: \[ \frac{d}{dy} g(h(y)) = g'(h(y)) \cdot h'(y) \]
In our case, for \( \text{ln}(x-y) \) with respect to \( y \), apply the chain rule:\[ \frac{d}{dy} \text{ln}(x-y) = \frac{1}{x-y} \, \cdot \, \frac{d}{dy} (x-y) \]
Since the derivative of \( x - y \) with respect to \( y \) is \( -1 \), we obtain:
\( f_y = x \cdot \left( \frac{1}{x-y} \cdot (-1) \right) = \frac{-x}{x-y} \).