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.
ScanSolve AI Answer
Step 1: Differentiate both sides of the equation (x^2y^3) + 3x^3y^2 = 6 with respect to x. (d)/(dx)[(x^2y^3) + 3x^3y^2] = (d)/(dx)[6] Step 2: Differentiate each term. For the first term, (d)/(dx)[(x^2y^3)]: Apply the chain rule. Let u = x^2y^3. Then (d)/(dx)[(u)] = (u) (du)/(dx). To find (du)/(dx) = (d)/(dx)[x^2y^3], apply the product rule: (d)/(dx)(x^2y^3) = (d)/(dx)(x^2) · y^3 + x^2 · (d)/(dx)(y^3) = (2x)y^3 + x^2(3y^2 (dy)/(dx)) = 2xy^3 + 3x^2y^2 (dy)/(dx) So, the derivative of the first term is: (x^2y^3) (2xy^3 + 3x^2y^2 (dy)/(dx)) = 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) For the second term, (d)/(dx)[3x^3y^2]: Apply the product rule: (d)/(dx)[3x^3y^2] = (d)/(dx)(3x^3) · y^2 + 3x^3 · (d)/(dx)(y^2) = (9x^2)y^2 + 3x^3(2y (dy)/(dx)) = 9x^2y^2 + 6x^3y (dy)/(dx) For the right side, (d)/(dx)[6]: The derivative of a constant is 0. (d)/(dx)[6] = 0 Step 3: Combine the differentiated terms into the equation. 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) + 9x^2y^2 + 6x^3y (dy)/(dx) = 0 Step 4: Group terms containing (dy)/(dx) on one side and other terms on the other side. 3x^2y^2 (x^2y^3) (dy)/(dx) + 6x^3y (dy)/(dx) = -2xy^3 (x^2y^3) - 9x^2y^2 Step 5: Factor out $dydx