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6.4 Derivatives in Two Dimensions: Directional Derivative and Partial Derivatives

Suppose we have a function of two variables, f ( x , y ) .

Such things are sometimes called scalar fields . (Scalar to indicate they are not vectors, and fields to indicate that there are two or more variables.)

We can choose a particular line in x y plane, (for example x = x 0 + r cos ⁡ θ , y = y 0 + r sin ⁡ θ ) and consider the function of r (with everything else in it fixed): f ( x 0 + r cos ⁡ θ , y 0 + r sin ⁡ θ ) .

d f d r is then called the directional derivative of f at ( x 0 , y 0 ) in the direction in the x y plane having slope tan ⁡ θ .

In other words we can, by picking out any particular line in the x y plane, reduce f to a function of a single value defined on that line and define the derivative of that one variable function with respect to distance on that line.

This derivative is called the directional derivative of f in the direction of the line. (You may examine the directional derivatives of functions of two variables in the applet .)

The directional derivative in the direction of the x-axis is called the partial derivative of f with respect to x , and is written as ∂ f ∂ x .

Similarly the directional derivative of f in the direction of the y-axis is called the partial derivative of f with respect to y , and is written as ∂ f ∂ y .

These partial derivatives are computable exactly as ordinary one dimensional derivatives are. When computing the partial derivative with respect to x , you treat y as a constant, and differentiate with respect to x exactly as you do in one dimension.

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