The Directional Derivative

Example Find the directional derivative of F(x,y), where F(x,y) is defined by

F(x, y) = 2x² + 3xy + 4 y² at the point, P(1,1) in the direction of the unit vector

u = ( i + j)/√2

Strategy: Calculate the gradient of F and take the dot product with u to determine

the directional derivative.

grad F = ∂F/∂x i + ∂F/∂y j = (4 x + 3 y) i + (3x + 8y) j

At point P (1,1) grad F = 7 i + 11 j

So dF / du = [ 7 i + 11 j ] ∙ [( i + j)/√2 ] = 18/√2 (result)

Example: From the previous example, F(x,y) = T(x,y) = 2x² + 3xy + 4 y² .

Suppose the function, T(x,y), represents temperature distribution.

Now find the maximum change of the temperature at point P(1, 1).

The maximum change will be given by the maximum directional derivative at point P

and will be in the direction of grad T.

Strategy: Calculate the unit vector in the direction of grad T and then take the

dot product with grad T.

grad T = (4 x + 3y ) i + (3x + 8y) j

At P(1, 1) (from above) grad T = 7 i + 11 j

So u = [ 7 i + 11 j ] / √ ( 7² + 11² ) = [ 7 i + 11 j ] / √ (170 )

So the maximum directional derivative = grad T ∙ u

= [7 i + 11 j ] ∙ [7 i + 11 j ] / √ (170 )]

Maximum change in temperature at P = (49 + 121) / √ (170 ) = √ (170 ) (result)

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