Question

x*y^2*z^3+x^3*y^2*z+1=x+y+z. Find dz/dx at (1, 1, 1).

Answer 1

what i like doing in these situations is just changing one of the variables to some number, because I know it will be treated as a constant. In this case I would make y some number since we're differentiating z wrt x. So rewrite this as:
(by the way any number would do, I just picked 5)
x*(5^2)*z^3 + x^3(5^2)z+1 = x + 5 + z

It's just something that I prefer doing because it reminds me that I need to treat y as a constant.

Answer 2

so now just use implicit differentiation:

(5^2) [x * z^3] + (5^2) [x^3 * z] + 1 = x + 5 + z

Whatever's in the brackets is going to use the product rule.

Answer 3

I just decided to change the 5s back to ys because I somewhat confused myself.
(y^2) [ x * 3z^2 (dz/dx) + z^3 ] + (y^2) [x^3 * (dz/dx) + z * 3x^2] = 1 + 0 + dz/dx

Answer 4

The 0 on the right hand side is coming from differentiating y - as far as dz/dx is concerned, y is a constant, it does not involve any z or x terms

Answer 5

So now just do the algebra and get dz/dx on its own.

Answer 6

Wait a minute, let me do the work.

Answer 7

Thank you so much, I've got the correct answer.

Answer 8

Good job