Partial Derivatives

A partial derivative measures how $f$ changes when one variable moves while the others are held fixed.

$$\frac{\partial f}{\partial x} = \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$$

Key insight: Partial derivatives reduce a multivariable problem to a single-variable one by treating all other variables as constants.

Notation

Geometric interpretation: $f_x(a,b)$ is the slope of $f$ at $(a,b)$ in the $x$-direction. Likewise, $f_y(a,b)$ is the slope in the $y$-direction.

Computation & Rules

Example: For $f(x,y) = x^{2}y + 3xy^{2}$:

$$f_x = 2xy + 3y^2, \qquad f_y = x^2 + 6xy$$

Clairaut's theorem: If $f_{xy}$ and $f_{yx}$ are both continuous, the order of differentiation does not matter:

$$f_{xy} = f_{yx}$$

Common pitfall:$\partial f / \partial x$ freezes every other variable, answering a deliberately narrow question. The real-world change of $f$ when several inputs move at once is the chain rule's job, not a single partial's.