Find partial derivative dz/dx and dz/dy on this function.

Question

The equation 30/x + 10/y + 5/z - 5 = 0 defines z as a function of x and y. Find the partial derivatives dz/dx and dz/dy.

Answer 1

6/x + 2/y + 5/z - 1 = 0

6/x + 2/y - 1 = 5/z

(6y + 2x - xy)/xy = 5/z

z = 5xy / (6y + 2x - xy)

Quotient Rule:

dz/dx = [(6y + 2x - xy) (5y) - (5xy)(2-y)]/ (6y+2x-xy)^2

= (5y) [6y +2x - xy - 10 + 5y]/(6y+2x-xy)^2

= (5y) [11y + 2x - xy - 10]/(6y+2x-xy)^2

dz/dx = [(6y + 2x - xy)(5x) - (5xy)(6 - y)]/(6y+2x-xy)^2

= (5x) [ 6y + 2x - xy - 6y + 5y]/(6y+2x-xy)^2

= (5x) [ 5y - xy + 2x]/(6y+2x-xy)^2