Partial derivatives need help! z= (all to the square root)x^2 - x+ y^2+1?

Question

z= (all to the square root)x^2 - x+ y^2+1

Answer 1

You find the partial derivatives by applying the chain rule:
z = √( f(x,y)) = √(x² + x + y² + 1)

→
∂z/∂x = ∂f/∂x · dz/df = (2x + 1) / [2·√(x² + x + y² + 1)]
and
∂z/∂y = ∂f/∂y · dz/df = (2y) / [2·√(x² + x + y² + 1)] = y / √(x² + x + y² + 1)

Answer 2

z = (x² - x + y² + 1)^(1/2)
ðz/ðx = (1/2).(x² - x + y² + 1)^(-1/2).(2x - 1)
ðz/ðy = (1/2).(x² - x + y² + 1)^(-1/2).(2y)