Question

What is the first derivative of `(y)*(sqrt(x)) - (x)*(sqrt(y)) = 16`?

Answer 1

`dy/dx = (y(2sqrtx-1))/(x(2sqrty-1))`

Explanation:

Differentiate both sides of the equation with respect to `x`:

`d/dx(ysqrtx-xsqrty) = 0`

`d/dx(ysqrtx) = d/dx(xsqrty)`

using the product rule:

`y/(2sqrtx)+ dy/dx sqrtx = x/(2sqrty)dy/dx + sqrty`

solving for `dy/dx`:

`dy/dx(sqrtx-x/(2sqrty)) = sqrty-y/(2sqrtx)`

`dy/dx((2sqrtxsqrty-x)/(2sqrty)) = (2sqrtxsqrty-y)/(2sqrtx)`

`dy/dx = ((2sqrtxsqrty-y)/(2sqrtx))((2sqrty)/(2sqrtxsqrty-x))`

`dy/dx = ((2ysqrtx-y)/(2xsqrty-x))`

`dy/dx = (y(2sqrtx-1))/(x(2sqrty-1))`