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 with respect to x. The given equation is (x^2y^3) + 3x^3y^2 = 6. (d)/(dx)[(x^2y^3) + 3x^3y^2] = (d)/(dx)[6] Step 2: Differentiate each term using the chain rule and product rule. For the first term, (d)/(dx)[(x^2y^3)]: Using the chain rule, (d)/(dx)[(u)] = (u) (du)/(dx), where u = x^2y^3. Now, find (du)/(dx) = (d)/(dx)[x^2y^3] using the product rule. Let f = x^2 and g = y^3. Then (df)/(dx) = 2x and (dg)/(dx) = 3y^2 (dy)/(dx). So, (d)/(dx)[x^2y^3] = (2x)(y^3) + (x^2)(3y^2 (dy)/(dx)) = 2xy^3 + 3x^2y^2 (dy)/(dx). Therefore, the derivative of the first term is: (x^2y^3)(2xy^3 + 3x^2y^2 (dy)/(dx)) For the second term, (d)/(dx)[3x^3y^2]: Using the product rule, let f = 3x^3 and g = y^2. Then (df)/(dx) = 9x^2 and (dg)/(dx) = 2y (dy)/(dx). So, the derivative of the second term is: (9x^2)(y^2) + (3x^3)(2y (dy)/(dx)) = 9x^2y^2 + 6x^3y (dy)/(dx) The derivative of the right side is (d)/(dx)[6] = 0. Step 3: Combine the differentiated terms. (x^2y^3)(2xy^3 + 3x^2y^2 (dy)/(dx)) + 9x^2y^2 + 6x^3y (dy)/(dx) = 0 Step 4: Expand the equation and group terms containing (dy)/(dx). 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) + 9x^2y^2 + 6x^3y (dy)/(dx) = 0 Move terms without (dy)/(dx) to the right 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 (dy)/(dx). (dy)/(dx) (3x^2y^2 (x^2y^3) + 6x^3y) = -2xy^3 (x^2y^3) - 9x^2y^2 Step 6: Solve for (dy)/(dx). (dy)/(dx) = (-2xy^3 (x^2y^3) - 9x^2y^2)/(3x^2y^2 (x^2y^3) + 6x^3y) Step 7: Simplify the expression by factoring out common terms from the numerator and denominator. Factor out -xy^2 from the numerator: -xy^2(2y (x^2y^3) + 9x) Factor out 3xy from the denominator: 3xy(xy (x^2y^3) + 2x^2) So, (dy)/(dx) = (-xy^2(2y (x^2y^3) + 9x))/(3xy(xy (x^2y^3) + 2x^2)) Cancel xy from the numerator and denominator (assuming x ≠ 0 and y ≠ 0): (dy)/(dx) = (-y(2y (x^2y^3) + 9x))/(3(xy (x^2y^3) + 2x^2)) The final answer is (dy)/(dx) = (-y(2y (x^2y^3) + 9x))/(3(xy (x^2y^3) + 2x^2)). That's 2 down. 3 left today — send the next one.