( ( e^( -2( X^(1/2) ) ) / x^(1/2) ) - (dx/dy) = 1

To solve the given equation:

( ( e^( -2( X^(1/2) ) ) / x^(1/2) ) - (dx/dy) = 1

we need to isolate the derivative term (dx/dy) and solve for it.

Step 1: Multiply through by x^(1/2) to remove the fraction:

e^(-2√x) - (dx/dy)(x^(1/2)) = x^(1/2)

Step 2: Add (dx/dy)(x^(1/2)) to both sides to isolate the derivative term:

e^(-2√x) = (dx/dy)(x^(1/2)) + x^(1/2)

Step 3: Factor out (dx/dy) on the right side:

e^(-2√x) = (dx/dy)(x^(1/2) + 1)

Step 4: Divide both sides by (x^(1/2) + 1) to solve for (dx/dy):

(dx/dy) = e^(-2√x) / (x^(1/2) + 1)

Therefore, the solution for the derivative term (dx/dy) is:

(dx/dy) = e^(-2√x) / (x^(1/2) + 1)