Question

if sin(-3x-2y+z)=0, use implicity differentiation to find the first partial derivatives dz/dx and dz/dy at the point (0,0,0)

Answer 1

To find the first partial derivatives dz/dx and dz/dy using implicit differentiation, differentiate the given equation with respect to x and y while treating z as a function of x and y. Both equations will give the same result: cos(-3x-2y+z) = 0. Therefore, dz/dx and dz/dy are zero at the point (0,0,0).

Step-by-step explanation:

To find the first partial derivatives dx/dz and dy/dz of the implicit function sin(−3x−2y+z)=0 at the point (0, 0, 0), we can use implicit differentiation.

Given: sin(−3x−2y+z)=0

- 3cos(−3x−2y+z)⋅ dx/dz =0

- 2cos(−3x−2y+z)⋅ dy/dz =0

Evaluate at (0, 0, 0):

dy/dz=0

So,

dx/dz=0 and dz/dy=0 at (0, 0, 0).