Question

Define the partial derivative of the function z

Answer 1

The partial derivative of a function \( z = f(x, y) \) with respect to one of its variables (either \( x \) or \( y \)) measures how the function \( z \) changes as the variable changes, while holding the other variable constant.

1. **Partial Derivative with Respect to \( x \)**:
The partial derivative of \( z \) with respect to \( x \) is denoted as \( \frac{\partial z}{\partial x} \) or \( f_x(x, y) \). It is defined as:
\[
\frac{\partial z}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}
\]
This expression gives the rate of change of \( z \) relative to \( x \) while keeping \( y \) fixed.

2. **Partial Derivative with Respect to \( y \)**:
Similarly, the partial derivative of \( z \) with respect to \( y \) is denoted as \( \frac{\partial z}{\partial y} \) or \( f_y(x, y) \). It is defined as:
\[
\frac{\partial z}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y}
\]
This expression gives the rate of change of \( z \) relative to \( y \), while keeping \( x \) constant.

In summary, the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) provide insight into how the function \( z \) varies with respect to each variable independently.