   Chapter 14.6, Problem 4E

Chapter
Section
Textbook Problem

Find the directional derivative of f at the given point in the direction indicated by the angle θ.4. f(x, y) = xy3 – x2, (1, 2), θ = π/3

To determine

To find: The directional derivative of the function f(x,y)=xy3x2 at the point (1,2) in the direction of the angle θ=π3 .

Explanation

Given:

The function is, f(x,y)=xy3x2 .

Result used:

The directional derivative of the function f(x,y) at f(x0,y0) in the direction of the angle θ is Duf(x,y)=fx(x,y)cosθ+fy(x,y)sinθ .

Calculation:

Substitute the respective values and obtain Duf(x,y) as follows,

Duf(x,y)=fx(x,y)cosθ+fy(x,y)sinθ=x(xy3x2)cos(π3)+y(xy3x2)sin(π3)=(y32x)(12)+(3xy2)(32)

Thus, the directional derivative of f(x,y)

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