Implicit differemation 5) \( \sqrt{x y}=\ln (x y) \)

We wish to differentiate   √(xy) = ln(xy) with respect to x, treating y as a function of x. Step 1. Write the equation in a clear form:   (1) √(xy) = ln(xy). Step 2. Differentiate both sides with respect to x. Notice that both sides involve the function u = xy. We have:  • The left‐hand side is √(xy) = [xy]^(1/2). By the chain rule,   d/dx { [xy]^(1/2) } = (1/(2√(xy))) · d/dx(xy).  • The right‐hand side is ln(xy). Again by the chain rule,   d/dx { ln(xy) } = (1/(xy)) · d/dx(xy). Step 3. Now compute d/dx(xy). Since x and y are both functions of x (with y = y(x)), use the product rule:   d/dx(xy) = y + x·y′. Step 4. Substitute these derivatives back into the original differentiations. Thus, the derivative of the left‐hand side is:   (1/(2√(xy)))(y + x y′), and the derivative of the right‐hand side is:   (1/(xy))(y + x y′). So our differentiated equation becomes:   (1/(2√(xy)))(y + x y′) = (1/(xy))(y + x y′). Step 5. Factor Out the Common Term Notice that (y + x y′) appears on both sides. Therefore we can write:   (y + x y′) · [1/(2√(xy)) – 1/(xy)] = 0. For a product to be zero, at least one of the factors must vanish. This gives us two cases: Case 1. Suppose   y + x y′ = 0. Then,   y′ = – y/x. Case 2. Suppose instead that   1/(2√(xy)) – 1/(xy) = 0. Solve for xy:   1/(2√(xy)) = 1/(xy). Multiply both sides by 2√(xy)·(xy) (assuming xy > 0 so that neither denominator is zero) to clear fractions:   xy = 2√(xy). Now, let u = √(xy) so that u^2 = xy. Then the equation becomes:   u^2 = 2u. This simplifies to:   u(u – 2) = 0. Since u = √(xy) must be positive (note that ln(xy) is only defined when xy > 0), we have u ≠ 0. Therefore, u – 2 = 0, or u = 2. Returning to the original variables:   √(xy) = 2 ⟹ xy = 4. Thus, if xy = 4 then the factor 1/(2√(xy)) – 1/(xy) vanishes. In this situation, the differentiated equation is automatically satisfied for any value of y′. In other words, no additional information about y′ is obtained when xy = 4. Summary of the Answer: The result of implicit differentiation is:   (1/(2√(xy)) – 1/(xy)) · (y + x y′) = 0. Thus, either:   y + x y′ = 0 ⟹ y′ = – y/x, or   xy = 4. Either answer (or an explanation that includes the two cases) is acceptable.