]> 6.4 Derivatives in Two Dimensions: Directional Derivative and Partial Derivatives

## 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.