Partial Derivatives

Definition

A partial derivative measures the rate of change of a multivariable function with respect to one variable while all other variables are held constant. $f_x(x, y) = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}$

• How to read: “The partial derivative of f with respect to x evaluated at x and y is equal to the limit as h approaches zero of the quantity f evaluated at x plus h and y, minus f evaluated at x and y, all divided by h.”
• Meaning: Rate of change of $f$ in the $x$-direction with $y$ held fixed—the multivariable analog of an ordinary derivative.

Why It Matters

We live in a multivariable world where nothing happens in isolation. Partial derivatives are the only tool we have for “freezing” the chaos of the world to see how one single factor (like interest rates or temperature) affects the whole. Without them, we cannot navigate “gradient” landscapes—we would be blind to the direction of steepest descent in a cost function or the direction of maximum heat flow, leading to sub-optimal decisions and system failures.

Core Concepts

• Notation: $f_x, \frac{\partial f}{\partial x}, z_x$.
  • How to read: “The partial derivative of f with respect to x, or the partial derivative of z with respect to x.”
  • Meaning: All three notations denote the partial derivative with respect to $x$.
• Geometric Interpretation: $f_x(x_0, y_0)$ is the slope of the tangent line to the surface $z = f(x, y)$ at the point $(x_0, y_0)$ in the direction of the positive $x$-axis.
  • How to read: “The partial derivative of f with respect to x evaluated at the point x zero and y zero.”
  • Meaning: Slice the surface with the plane $y = y_0$; the partial derivative is the slope of that trace curve at $x_0$.
• Computation: Treat all variables except the one being differentiated as constants.

Connected Concepts

• Second-Order Partial Derivatives
• Function Definition, Function Notation
• Partial Fraction Decomposition