find dx/dy implicitly from √(x+y) = 1 + (x^2)(y^2). then solve for dx/dy explicitly.
To find dx/dy implicitly, we can start by differentiating both sides of the given equation with respect to y. Remember to use the chain rule when necessary.
Starting with the left-hand side:
Let u = x + y
Using the chain rule, the derivative of √(x+y) with respect to y can be found by differentiating u with respect to y and applying the power rule:
d/dy[√(x+y)] = d/du[√u] * du/dy
Applying the chain rule, d/du[√u] = 1/(2√u)
And du/dy = d/dy[x+y] = 1 + dx/dy
So, the left-hand side becomes:
1/(2√(x+y)) * (1 + dx/dy)
Moving on to the right-hand side:
Differentiating with respect to y, we have:
d/dy[1 + (x^2)(y^2)] = 0 + d/dy[(x^2)(y^2)]
= (2x^2y)(dy/dy) + (x^2)(2y)(dy/dy)
= 2x^2y + 2xy^2(dx/dy)
Now, equating the left-hand side and right-hand side derivatives, we get:
1/(2√(x+y)) * (1 + dx/dy) = 2x^2y + 2xy^2(dx/dy)
Next, we can isolate dx/dy to solve for it explicitly:
1 + dx/dy = 4xy^2(dx/dy) + 4x^2y√(x+y)
Rearranging the terms and factoring out dx/dy:
(1 - 4xy^2)dx/dy = 4x^2y√(x+y) - 1
Dividing both sides by (1 - 4xy^2):
dx/dy = (4x^2y√(x+y) - 1)/(1 - 4xy^2)
Now, we have the explicit expression for dx/dy in terms of x and y.