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^2y^2 = 6. We need to apply the chain rule and product rule for differentiation. Remember that y is a function of x, so (d)/(dx)(y^n) = ny^n-1(dy)/(dx). Differentiating the first term, (x^2y^3): Using the chain rule, (d)/(dx)((u)) = (u) (du)/(dx), where u = x^2y^3. To find (du)/(dx) = (d)/(dx)(x^2y^3), we use the product rule: (d)/(dx)(fg) = f'g + fg'. Let f = x^2 and g = y^3. f' = (d)/(dx)(x^2) = 2x. g' = (d)/(dx)(y^3) = 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, (d)/(dx)((x^2y^3)) = (x^2y^3)(2xy^3 + 3x^2y^2 (dy)/(dx)). Differentiating the second term, 3x^2y^2: Using the product rule for x^2y^2: Let f = x^2 and g = y^2. f' = (d)/(dx)(x^2) = 2x. g' = (d)/(dx)(y^2) = 2y (dy)/(dx). So, (d)/(dx)(3x^2y^2) = 3 ( (2x)(y^2) + (x^2)(2y (dy)/(dx)) ) = 3(2xy^2 + 2x^2y (dy)/(dx)) = 6xy^2 + 6x^2y (dy)/(dx). Differentiating the constant term, 6: (d)/(dx)(6) = 0. Step 2: Combine the differentiated terms and solve for (dy)/(dx). (x^2y^3)(2xy^3 + 3x^2y^2 (dy)/(dx)) + 6xy^2 + 6x^2y (dy)/(dx) = 0 Distribute (x^2y^3): 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) + 6xy^2 + 6x^2y (dy)/(dx) = 0 Group terms containing (dy)/(dx) on one side and other terms on the other side: 3x^2y^2 (x^2y^3) (dy)/(dx) + 6x^2y (dy)/(dx) = -2xy^3 (x^2y^3) - 6xy^2 Factor out (dy)/(dx): (dy)/(dx) (3x^2y^2 (x^2y^3) + 6x^2y) = -2xy^3 (x^2y^3) - 6xy^2 Isolate (dy)/(dx): (dy)/(dx) = (-2xy^3 (x^2y^3) - 6xy^2)/(3x^2y^2 (x^2y^3) + 6x^2y) Step 3: Simplify the expression for (dy)/(dx). Factor out common terms from the numerator and the denominator. Numerator: -2xy^2(y (x^2y^3) + 3) Denominator: 3x^2y(y (x^2y^3) + 2) (dy)/(dx) = (-2xy^2(y (x^2y^3) + 3))/(3x^2y(y (x^2y^3) + 2)) Cancel common factors x and y: (dy)/(dx) = (-2y(y (x^2y^3) + 3))/(3x(y (x^2y^3) + 2)) The final answer is (dy)/(dx) = (-2y(y (x^2y^3) + 3))/(3x(y (x^2y^3) + 2)). That's 2 down. 3 left today — send the next one.