Question

I'm having troubles finding tangent planes when you need to differentiate implicitly. For example, how would you find the tangent plane of arctan(yz)=xz^y at (pi/4, 1, 1)?

Answer 1

Professor Auroux gives another great method which could skip the implicit differentiation totally.
Steps:
1) Set a new 3-var function G(x,y,z) = arctan(yz) - xz^y = 0.
2) Find each partial derivative, Gx, Gy, Gz.
3) Write the exact(total) differential for G : dG = Gx*dx + Gy*dy + Gz*dz . Since G = 0 which is a constant, so dG = Gx*dx + Gy*dy + Gz*dz = 0
4) Write down the equation of the tangent plane according to this exact differential formula : Gx*dx + Gy*dy + Gz*dz = 0
, which is:

" Gx(pi/4, 1, 1)*(x-pi/4) + Gy(pi/4, 1, 1)*(y-1) + Gz(pi/4, 1, 1)*(z-1) = 0 "
Done.