220k views
3 votes
Suppose z equals f (x comma y ), where x (u comma v )space equals space 2 u plus space v squared, y (u comma v )space equals space 3 u minus v, f subscript x (6 comma 1 )equals 3, and f subscript y (6 comma 1 )equals negative 1. Evaluate fraction numerator partial differential z over denominator partial differential v end fraction at (u comma v )equals (1 comma 2 ). **Note 1: Your answer will be an integer.

User Hrishikesh
by
7.7k points

1 Answer

3 votes


z=f(x(u,v),y(u,v)),\begin{cases}x(u,v)=2u+v^2\\y(u,v)=3u-v\end{cases}

We're given that
f_x(6,1)=3 and
f_y(6,1)=-1, and want to find
(\partial z)/(\partial v)(1,2).

By the chain rule, we have


(\partial z)/(\partial v)=(\partial z)/(\partial x)(\partial x)/(\partial v)+(\partial z)/(\partial y)(\partial y)/(\partial v)

and


(\partial x)/(\partial v)=2v


(\partial y)/(\partial v)=-1

Then


(\partial z)/(\partial v)(1,2)=(\partial z)/(\partial x)(6,1)(\partial x)/(\partial v)(1,2)+(\partial z)/(\partial y)(6,1)(\partial y)/(\partial v)(1,2)

(because the point
(x,y)=(6,1) corresponds to
(u,v)=(1,2))


\implies(\partial z)/(\partial v)(1,2)=3\cdot2\cdot2+(-1)\cdot(-1)=\boxed{13}

User Bottleboot
by
8.5k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories