The Directional Derivative, its Definition and Physical Interpretation

In Nut Shell: The simple derivative of a function, y, of one independent variable, x,

can be viewed as giving the derivative of y in the direction of x. This same concept

extends to functions of more than one independent variable and any given direction.

The directional derivative of a function, D_uf(x,y), gives the rate of change of f(x,y) on a

line through the point, P, in the direction of the unit vector, u. See the figure below.

∂/∂x represents the change of the function (slope) in the x-direction,

∂/∂y represents the change of the function (slope) in the y-direction,

and D_u f represents the change of the function (slope) through P in the direction of

the unit vector u.

One can show that the directional derivative of f(x,y) in the direction of the unit vector, u,

D_u f(x,y) , is the dot product of the gradient of f(x,y) with the unit vector, u. It provides a

convenient method to calculate the directional derivative.

D_u f(x,y) = grad f(x,y) _∙ u

Recall that the gradient function, grad f(x,y), points in the direction in which the function

f(x,y) increases (or decreases) most rapidly and the dot product with u gives the

component of grad f(x,y) in the direction of u .

The maximum value of the directional derivative is in the direction of grad f(x,y). So it

occurs when u is a unit vector in the direction of grad f .

So u = grad f / | grad f |

i.e. If f (x ) represents the temperature, then proceeding in the direction of u

gives the greatest temperature change.

Click here for more discussion of the directional derivative.

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