Use implicit differentiation to find dz/dx and dz/dy for 3x^2*z+2z^3-3yz=0.
Answer: dz/dx=-6xz/(3x^2+6z^2-3y)
dz/dy=3z/(3x^2+6z^2-3y)
This is Calculus 3 topic. Show your work through steps.
Use implicit differentiation to find dz/dx and dz/dy for 3x^2*z+2z^3-3yz=0.
Answer: dz/dx=-6xz/(3x^2+6z^2-3y)
dz/dy=3z/(3x^2+6z^2-3y)
This is Calculus 3 topic. Show your work through steps.
Sun,
Hello :) Let's take a look at this problem.
3zx^2 + 2z^3 - 3yz = 0
Let's start by moving the 3yz over:
3zx^2 + 2z^3 = 3yz
Implicit differentiation is really just differentiating without making an explicit statement. In other words, we will leave this equation "as is" instead of restating to say that x = "this" or y = "that". Let's differentiate for dz/dx:
d/dx (3zx^2) + d/dx (2z^3) = d/dx (3yz)
3 d/dx (zx^2) + 2 d/dx (z^3) = 3 d/dx (yz)
Use product rule ( (uv)' = uv' + u'v ) for d/dx zx^2. For d/dx(yz), remember that when using implicit differentiation, if differentiating with respect to x (finding dz/dx), treat y as a constant.
3(z * 2x + dz/dx * x^2) + 2(3z^2)(dz/dx) = 3y * d/dx z
3( 2xz + x^2(dz/dx) ) + 6z^2(dz/dx) = 3y(dz/dx)
6xz + 3x^2(dz/dx) + 6z^2(dz/dx) = 3y(dz/dx) (Let's round up the dz/dx terms)
3x^2(dz/dx) + 6z^2(dz/dx) - 3y(dz/dz) = -6xz
dz/dx (3x^2 + 6z^2 - 3y) = -6xz
dz/dx = -6xz / (3x^2 + 6z^2 - 3y)
As was to be shown. The process is the same of course for determining dz/dy. Again, remember that since we are now differentiating with respect to y, we will treat x as a constant.
Find dz/dy for 3zx^2 + 2z^3 - 3yz = 0
3zx^2 + 2z^3 = 3yz
d/dy 3zx^2 + d/dy 2z^3 = d/dy 3yz
3 d/dy zx^2 + 2 d/dy z^3 = 3 d/dy yz (Now treat x as a constant when differentiating, use product rule for d/dy yz)
3x^2 d/dy z + 2(3z^2)(dz/dy) = 3(y(dz/dy) + 1(z))
3x^2(dz/dy) + 6z^2(dz/dy) = 3y(dz/dy) + 3z (Round up dz/dy)
3x^2(dz/dy) + 6z^2(dz/dy) - 3y(dz/dy) = 3z
dz/dy ( 3x^2 + 6z^2 - 3y ) = 3z
dz/dy = 3z / ( 3x^2 + 6z^2 - 3y )
As was to be shown! Hope this helps :)
-Skyler
There's no need to move the "3yz" term to the right-hand side (as you did before you started differentiating).
You're absolutely right. It also doesn't hurt anything :) I did it to satisfy my personal obsessive compulsive desire to eliminate that lone 0.
Comments
Thank you so much for the help. Can you please answer my other question about the gradient?