find dx/dy implicitly from √(x+y) = 1 + (x^2)(y^2). then solve for dx/dy explicitly.
To find dx/dy implicitly from the equation √(x+y) = 1 + (x^2)(y^2), we will differentiate both sides of the equation with respect to x and then solve for dx/dy. Here's how to proceed:
Step 1: Differentiate both sides of the equation with respect to x using the chain rule. Remember that y is a function of x, so when we differentiate y with respect to x, we use the chain rule.
On the left side, we have √(x+y). Applying the chain rule, we get:
d/dx (√(x+y)) = d/dx (x+y)^(1/2)
= (1/2)(x+y)^(-1/2) * d/dx (x+y)
On the right side, we have 1 + (x^2)(y^2), which can be rewritten as 1 + x^2y^2.
Step 2: Set the resulting derivatives equal to each other and solve for dy/dx.
(1/2)(x+y)^(-1/2) * d/dx (x+y) = d/dx (1 + x^2y^2)
Step 3: Simplify and solve for dy/dx.
(1/2)(x+y)^(-1/2) * (1 + dy/dx) = 2xy^2
Now, multiply both sides by 2(x+y)^(1/2):
(x+y)^(1/2) * (1 + dy/dx) = 4xy^2
Expand the left side:
(x+y)^(1/2) + dy/dx * (x+y)^(1/2) = 4xy^2
Subtract (x+y)^(1/2) from both sides:
dy/dx * (x+y)^(1/2) = 4xy^2 - (x+y)^(1/2)
Finally, solve for dy/dx:
dy/dx = (4xy^2 - (x+y)^(1/2))/(x+y)^(1/2)
This is the implicit expression for dx/dy.
To find dx/dy explicitly, you can solve the equation above for dy/dx. Rearrange the terms and simplify if possible.