Partial Derivatives

When a function has more than one input, "the derivative" is ambiguous until you say which input is changing. A partial derivative answers that: differentiate with respect to one variable and treat the rest as constants. For $f(x,y)$ the two first partials are

The symbol $\frac{\partial f}{\partial x}$ means differentiate with respect to $x$ while $y$ stays fixed, and $\frac{\partial f}{\partial y}$ does the same with respect to $y$ while $x$ stays fixed.

When to use this method

Use partial derivatives whenever an output depends on several inputs and you want the rate of change in one direction at a time. If temperature is modeled by $T(x,y)$, then $\frac{\partial T}{\partial x}$ measures how temperature changes as you move in the $x$ direction while staying at the same $y$. That "same $y$ value" condition is the entire idea, and it is what distinguishes a partial derivative from an ordinary one-variable derivative.

The steps

1. Choose the variable. Decide whether you want the rate of change with respect to $x$, $y$, or another variable.
2. Hold the others fixed. Treat every other variable as a constant for that derivative.
3. Differentiate normally. Apply the usual derivative rules to the chosen variable only.
4. Evaluate if needed. Simplify first, then substitute a point only after you have the derivative formula.

The whole procedure on one example

Let

Find $f_x$ (hold $y$ constant). Then $x^2y$ behaves like a constant times $x^2$, and $3y^2$ is a constant with respect to $x$:

Find $f_y$ (hold $x$ constant). Now $x^2y$ behaves like $x^2 \cdot y$ with $x^2$ a constant multiplier, and $3y^2$ differentiates normally:

Evaluate at $(1,2)$, after differentiating:

The pattern: the variable you are not using behaves like a number during that derivative. That is also why

since $3y^2$ does not change as $x$ changes when $y$ is held fixed.

Where each step tends to stall, and how to check

Step 2 (hold others fixed): the most common failure is letting $y$ drift while differentiating in $x$. Self-check by confirming every term with no chosen variable became a constant, so its derivative is $0$.

Step 1 (choose the variable): $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ answer different questions; label which one the problem wants before you start.

Step 4 (evaluate): plugging in a point before differentiating hides the structure of the function, so always differentiate first. And do not assume partials exist everywhere — they can fail at points where the function is not well behaved.

A mental picture helps: think of $z=f(x,y)$ as a surface. $\frac{\partial f}{\partial x}$ is the slope of a slice taken in the direction where $y$ is fixed; $\frac{\partial f}{\partial y}$ is the slope of the slice where $x$ is fixed.

Run the full routine on

Find $g_x$ and $g_y$, then evaluate both at $(2,1)$, checking after each that you truly held the other variable constant. Partial derivatives appear throughout multivariable calculus — gradients, tangent planes, optimization, differential equations — and in physics, economics, and engineering, always asking the same practical question: what happens if one input changes while the others stay fixed?