Find dy/dx by implicit differentiation for the following

Question

sqrt xy = x^5y+52

Answer 1

We differentiate both sides: 1/2(xy)^-1/2(y + xy') = 5x⁴y + x⁵y'; 1/2x^-1/2y^1/2 + 1/2x^1/2y^-1/2y' = 5x⁴y + x⁵y';

(1/2x^1/2y^-1/2 - x⁵)y' = 5x⁴y - 1/2x^1/2y^-1/2;

y' = dy/dx = (10x⁴y - x^1/2y^-1/2)/(x^1/2y^-1/2 - 2x⁵)