\( \frac { \partial z } { \partial x } = \lim _ { h \rightarrow 0 } \frac { z ( x + h , y ) - z ( x , y ) } { h } \)

You've provided the definition of the **partial derivative** of a function \( z = z(x, y) \) with respect to the variable \( x \). Let's break down and explain each component of the expression: \[ \frac{\partial z}{\partial x} = \lim_{h \rightarrow 0} \frac{z(x + h, y) - z(x, y)}{h} \] ### Components Explained: 1. **Partial Derivative (\( \frac{\partial z}{\partial x} \))**: - **Definition**: It measures how the function \( z(x, y) \) changes as only the variable \( x \) changes, while keeping \( y \) constant. - **Interpretation**: It tells you the rate at which \( z \) increases or decreases as \( x \) increases, assuming \( y \) does not change. 2. **Limit Expression (\( \lim_{h \rightarrow 0} \frac{z(x + h, y) - z(x, y)}{h} \))**: - **Numerator (\( z(x + h, y) - z(x, y) \))**: - Represents the change in the function \( z \) when \( x \) is increased by a small amount \( h \), while \( y \) remains the same. - **Denominator (\( h \))**: - The small increment by which \( x \) is increased. - **Limit as \( h \rightarrow 0 \)**: - Ensures that we're looking at the instantaneous rate of change as the increment \( h \) becomes infinitesimally small. ### Interpretation: The partial derivative \( \frac{\partial z}{\partial x} \) essentially captures the slope of the function \( z(x, y) \) in the direction of the \( x \)-axis. It tells you how sensitive the function \( z \) is to small changes in \( x \), holding \( y \) constant. ### Example: Suppose \( z(x, y) = x^2 y + \sin(x) \). Let's find \( \frac{\partial z}{\partial x} \) using the limit definition. 1. **Compute \( z(x + h, y) \)**: \[ z(x + h, y) = (x + h)^2 y + \sin(x + h) \] 2. **Subtract \( z(x, y) \)**: \[ z(x + h, y) - z(x, y) = [(x + h)^2 y + \sin(x + h)] - [x^2 y + \sin(x)] = 2xhy + h^2 y + \sin(x + h) - \sin(x) \] 3. **Divide by \( h \)**: \[ \frac{z(x + h, y) - z(x, y)}{h} = 2xy + h y + \frac{\sin(x + h) - \sin(x)}{h} \] 4. **Take the limit as \( h \rightarrow 0 \)**: \[ \frac{\partial z}{\partial x} = \lim_{h \rightarrow 0} \left(2xy + h y + \frac{\sin(x + h) - \sin(x)}{h}\right) = 2xy + 0 + \cos(x) = 2xy + \cos(x) \] ### Conclusion: The expression you've provided is the foundational definition for computing the partial derivative of a function with respect to one of its variables. Understanding this concept is crucial in fields like calculus, physics, engineering, and any area that involves multivariable functions.