A multivariable function can change in several independent input directions. A partial derivative measures change with respect to one variable while the others are held fixed.

For $f(x,y)$, the first partial derivatives are written

$$\frac{\partial f}{\partial x},\qquad \frac{\partial f}{\partial y}.$$

Differentiate one variable at a time

Consider

$$f(x,y)=x^2y+3y^2.$$

Holding $y$ constant,

$$\frac{\partial f}{\partial x}=2xy.$$

Holding $x$ constant,

$$\frac{\partial f}{\partial y}=x^2+6y.$$

The ordinary single-variable differentiation rules apply; the only change is deciding which variables are treated as constants.

Geometric meaning

For a surface $z=f(x,y)$, fixing $y=b$ gives a curve

$$z=f(x,b).$$

The value $f_x(a,b)$ is the tangent slope of this curve at $x=a$. Similarly, $f_y(a,b)$ measures the slope in the $y$ direction.

Higher partial derivatives

Partial derivatives can themselves be differentiated. For example,

$$f_{xx}=\frac{\partial^2f}{\partial x^2},\qquad f_{xy}=\frac{\partial^2f}{\partial y,\partial x}.$$

For sufficiently smooth functions, the mixed partial derivatives agree:

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

Partial derivatives describe coordinate-direction change, but they do not by themselves guarantee that a function has one coherent linear approximation in all directions.