Find dy/dx by implicit differentiation. √(5xy) = 6 + (x^2)y

Answer 1

First, we will take the derivative with respect to x on each side of the equation:

d/dx(5xy)^1/2 = d/dx(6 + x²y)

5^1/2(x + xy')/(2(xy)^1/2) = 2xy + x²y'

Now, we just need to do some algebra to solve for y':

5^1/2(y + xy') = 4(xy)^3/2 + 2x^5/2y^1/2y'

We'll bring everything multiplied by a y' to the left, and everything else to the right:

5^1/2xy' - 2x^5/2y^1/2y' = 4(xy)^3/2 - 5^1/2y

(5^1/2x - 2x^5/2y^1/2)y' = 4(xy)^3/2 - 5^1/2y

Solving for y':

y' = (4(xy)^3/2 - 5^1/2y)/(5^1/2x - 2x^5/2y^1/2)

You can do a few more simplifications to make the answer look a bit cleaner, but as far as finding the derivative goes, we've solved the problem!

Hope this helps!