Find dy/dx for x^(1/2)+y^(1/2)=4?

Answer 1

1/2x^(-1/2) + 1/2y^(-1/2) dy/dx = 0

dy/dx = -1/(2 sqrt(x)) / [1/(2 sqrt(y))]
=
- sqrt(y/x)

Answer 2

(1/2)x^(-1/2) + (1/2)y^(-1/2)*dy/dx = 0
x^(-1/2) + y^(-1/2)*dy/dx = 0
dy/dx = -x^(-1/2)/y^(-1/2) = y^(1/2)/x^(1/2) = sqrt(y)/sqrt(x)

This is implicit differentiation, where you take the derivative of both sides of the equation and then find the value of dy/dx algebraically. Just remember that d(f(y))/dx = (d(f(y))/dy)*(dy/dx), an application of the chain rule.

Answer 3

Differentiate this implicitely:
1/2 x^(-1/2) + 1/2 y^(-1/2) dy/dx = 0
So dy/dx = - (x/y)^(-1/2) = - (y/x) ^1/2

You can then substitute y^1/2 = 4 - x^1/2

So dy/dx = -[(4 - x^1/2) / x^1/2]