The Partial Derivative

No single slope

On a curve y = f(x) there is one slope at each point, the derivative. On a surface z = f(x, y) there is no single slope: standing at one point, the ground may rise steeply one way, gently another way, and fall a third way.

Take $f\left(x,y\right)=x^{2}y$f(x, y) = x²y at the point (1, 2), where the height is 1 × 2 = 2. A step of 0.1 in the x direction, to (1.1, 2), raises the height to 2.42. A step of 0.1 in the y direction, to (1, 2.1), raises it only to 2.1. The two directions climb at different rates.

Hold y constant

To measure the slope in the x direction, walk only in that direction. Then y does not change: it stays at 2. Putting y = 2 into f gives $z=2x^2$z = 2x², an ordinary function of one variable.

Its graph is the slice of the surface cut by the plane y = 2, a parabola. Its gradient is found by ordinary differentiation: $\frac{d}{dx}$d/dx of $2x^2$2x² is 4x, which is 4 at x = 1. So at (1, 2) the surface rises 4 units for each unit moved in the x direction.

The slice of $z=x^{2}y$z = x²y at y = 2, which is the parabola $z=2x^2$z = 2x², with its tangent at the point (1, 2). The tangent z = 4x − 2 has gradient 4: the slope of the surface in the x direction at that point.

The partial derivative

That slope is the partial derivative of f with respect to x, written $\frac{\partial f}{\partial x}$∂f/∂x. The curly $\partial$∂ is used in place of d to show that f depends on more than one variable and that all but one are held constant.

As a limit, $\frac{\partial f}{\partial x}$∂f/∂x at (a, b) is found from a step h in the x direction: take the change in height f(a + h, b) − f(a, b), divide it by h, and find the limit as h tends to 0. Only x moves; y stays at b. For $x^{2}y$x²y at (1, 2), steps of 0.1, 0.01 and 0.001 give the quotients 4.2, 4.02 and 4.002, which tend to 4.

In practice no limit is needed: treat y as a constant and differentiate in x by the ordinary rules. In $x^{2}y$x²y the y is a constant factor, so it stays, and $x^2$x² becomes 2x. So $\frac{\partial f}{\partial x}=2xy$∂f/∂x = 2xy, which is 2 × 1 × 2 = 4 at (1, 2).

The other direction

The partial derivative with respect to y, $\frac{\partial f}{\partial y}$∂f/∂y, holds x constant and differentiates in y. In $x^{2}y$x²y the $x^2$x² is now the constant factor and y differentiates to 1, so $\frac{\partial f}{\partial y}=x^2$∂f/∂y = x², which is 1 at (1, 2).

The slice at x = 1 is z = y, a straight line: a step h in the y direction changes the height by exactly h, so the difference quotient is exactly 1 for every step. At (1, 2) the surface rises 4 per unit in the x direction and 1 per unit in the y direction.

For $f\left(x,y\right)=x^{2}y+5y^2$f(x, y) = x²y + 5y², the term $5y^2$5y² has no x in it, so under $\frac{\partial }{\partial x}$∂/∂x it is a constant and contributes 0: $\frac{\partial f}{\partial x}=2xy$∂f/∂x = 2xy. Under $\frac{\partial }{\partial y}$∂/∂y it contributes 10y, so $\frac{\partial f}{\partial y}=x^2+10y$∂f/∂y = x² + 10y. At (1, 2) the slopes are 4 and 1 + 20 = 21.

The surface $f=x^2+2y^2$f = x² + 2y²: its contour map on the left, and on the right the slice through the point (1.5, 0.5). Holding y constant, the slice is $z=x^2+0.5$z = x² + 0.5, and its gradient at x = 1.5 is $\frac{\partial f}{\partial x}=2x=3$∂f/∂x = 2x = 3. Holding x constant instead, the slice is $z=2.25+2y^2$z = 2.25 + 2y², and its gradient is $\frac{\partial f}{\partial y}=4y=2$∂f/∂y = 4y = 2.

The usual mistakes

Treating y as a variable while differentiating in x. Using the product rule on $x^{2}y$x²y in x gives $2xy+x^2\frac{dy}{dx}$2xy + x² dy/dx, but y is held constant, so the second term is not there: $\frac{\partial f}{\partial x}=2xy$∂f/∂x = 2xy.

Deleting the constant instead of keeping it. Under $\frac{\partial }{\partial x}$∂/∂x the y in $x^{2}y$x²y is a constant factor, and a constant factor stays in the derivative, so the answer is 2xy, not 2x.

Differentiating a term with no x in it. Under $\frac{\partial }{\partial x}$∂/∂x, the term $5y^2$5y² is a constant, so it gives 0, not 10y.

Air under a piston

In the application below, the pressure of the air in a cylinder depends on its temperature T and its volume V. Holding V constant gives the rate at which the pressure rises with the temperature, and holding T constant gives the rate at which it falls as the volume grows.