Find the directional derivative of f=x^2 * y * z^3 along the curve x = e^(-u); y = 2 sin u + 1; z = u - cos u at the point P where u = 1

Salvador Whitehead

Salvador Whitehead

Answered question

2022-11-24

Find the directional derivative of f = x 2 · y · z 3 along the curve x = e u ;         y = 2 sin u + 1 ;         z = u cos u at the point P where u = 0
My working:
At u=0, x=1, y=1, z=-1 so let u = (1,1,-1).
Know that D u f ( x ) = f ( x ) · u = ( 2 x · y · z 3 , x 2 · z 3 , 3 x 2 · y · z 3 ) · ( 1 , 1 , 1 ) = ( 2 x · y · z 3 , x 2 · z 3 , 3 x 2 · y · z 2 )
At u=0, x=(1,1,-1) so D u f ( x ) = ( 2 · 1 · 1 · 1 , 1 · 1 , 3 · 1 · 1 · 1 ) = ( 2 , 1 , 3 )
However, I'm not sure if this is correct as I don't know whether I'm meant to substitute u into f to find the derivative at a specific point or not?

Answer & Explanation

Lukas Arias

Lukas Arias

Beginner2022-11-25Added 6 answers

Hint:
lim h 0 f ( x ( u + h ) , y ( u + h ) , z ( u + h ) ) f ( x ( u ) , y ( u ) , z ( u ) ) h = lim h 0 ( f x d x d u + f y d y d u + f z d z d u ) h h = f ( d x d u , d y d u , d z d u ) .

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