Find-the-directional-derivative-of-f-x-y-4x-3-3x-2-y-2-in-the-direction-given-by-the-angle-pi-3-and-also-Evaluate-directional-derivatives-at-the-point-1-2-

Question Number 210078 by Spillover last updated on 29/Jul/24

F i n d t h e d i r e c t i o n a l d e r i v a t i v e o f

f (x, y) = 4 x^{3} - 3 x^{2} y^{2} i n t h e d i r e c t i o n g i v e n

b y t h e a n g l e θ = \frac{π}{3}

a n d a l s o E v a l u a t e d i r e c t i o n a l d e r i v a t i v e s

a t t h e p o i n t (1, 2)

Answered by Spillover last updated on 30/Jul/24

D_{v} f (x, y) = f_{x} (x, y) a + f_{y} (x, y) b

f_{x} = 12 x^{2} - 3 y^{2}

f_{y} = - 6 x y

v = (a, b) = (\cos θ, \sin θ)

v = (a, b) = (\cos \frac{π}{3}, \sin \frac{π}{3}) = (\frac{1}{2}, \frac{\sqrt{3}}{2})

D_{v} f (x, y) = f_{x} (x, y) a + f_{y} (x, y) b

= (12 x^{2} - 3 y^{2}) (\frac{1}{2}) + (- 6 x y) (\frac{\sqrt{3}}{2})

D_{v} f (x, y) = 6 x^{2} - \frac{3}{2} y^{2} - 3 \sqrt{3} x y

D_{v} f (1, 2) = 6 (1^{2}) - \frac{3}{2} (2^{2}) - 3 \sqrt{3} 2 \times 1 = - 6 \sqrt{3}