If 2ˣ + 2ʸ = 2ˣ⁺ʸ, then dy/dx is:

If 2ˣ + 2ʸ = 2ˣ⁺ʸ, then dy/dx is:

• A. 2^(y−x)
• B. 2^(x−y)
• C. −2^(y−x)
• D. −2^(x−y)

Answer: C) −2^(y−x)

Explanation: Differentiating: 2ˣ ln 2 + 2ʸ ln 2 y′ = 2ˣ⁺ʸ ln 2 (1+y′). Divide by ln2: 2ˣ + 2ʸ y′ = 2ˣ⁺ʸ + 2ˣ⁺ʸ y′. Since 2ˣ⁺ʸ = 2ˣ + 2ʸ, substitute: 2ˣ + 2ʸ y′ = (2ˣ+2ʸ)(1+y′). Solve: y′ = −2ˣ/2ʸ = −2^(x−y). Actually solving: 2ˣ + 2ʸ y′ = 2ˣ+2ʸ + (2ˣ+2ʸ)y′ → 0 = 2ʸ + 2ˣ y′ → y′ = −2ʸ/2ˣ = −2^(y−x). But 2ˣ⁺ʸ = 2ˣ+2ʸ. So RHS = (2ˣ+2ʸ) + (2ˣ+2ʸ)y′ = 2ˣ+2ʸ + 2ˣ y′ + 2ʸ y′. LHS = 2ˣ + 2ʸ y′. Cancel 2ˣ and 2ʸ y′: 0 = 2ʸ + 2ˣ y′ → y′ = −2ʸ/2ˣ = −2^(y−x). So option 2.