This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Okay, here's a question on implicit differentiation and its solution. Problem: Find (dy)/(dx) for the equation x^3 + y^3 = 6xy. Step 1: Differentiate both sides with respect to x. Remember to apply the chain rule for terms involving y and the product rule for 6xy. (d)/(dx)(x^3 + y^3) = (d)/(dx)(6xy) (d)/(dx)(x^3) + (d)/(dx)(y^3) = 6(d)/(dx)(xy) 3x^2 + 3y^2 (dy)/(dx) = 6(1 · y + x · (dy)/(dx)) 3x^2 + 3y^2 (dy)/(dx) = 6y + 6x (dy)/(dx) Step 2: Gather terms containing (dy)/(dx) on one side. Move all terms with (dy)/(dx) to the left side and all other terms to the right side. 3y^2 (dy)/(dx) - 6x (dy)/(dx) = 6y - 3x^2 Step 3: Factor out (dy)/(dx). (dy)/(dx)(3y^2 - 6x) = 6y - 3x^2 Step 4: Solve for (dy)/(dx). Divide both sides by (3y^2 - 6x). (dy)/(dx) = (6y - 3x^2)/(3y^2 - 6x) We can simplify the expression by dividing the numerator and denominator by 3. (dy)/(dx) = (3(2y - x^2))/(3(y^2 - 2x)) (dy)/(dx) = (2y - x^2)/(y^2 - 2x) The derivative is (dy)/(dx) = (2y - x^2)/(y^2 - 2x). That's 3 down. 2 left today — send the next one.