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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To find (dy)/(dx) for the given equation (x^2 y^3) + 3x^3 y^2 = 6, we will use implicit differentiation. We differentiate both sides of the equation with respect to x, treating y as a function of x. Step 1: Differentiate each term with respect to x. (d)/(dx)((x^2 y^3)) + (d)/(dx)(3x^3 y^2) = (d)/(dx)(6) Step 2: Differentiate (x^2 y^3) using the chain rule. Let u = x^2 y^3. Then (d)/(dx)((u)) = (u) (du)/(dx). To find (du)/(dx) = (d)/(dx)(x^2 y^3), we use the product rule: (d)/(dx)(fg) = f'g + fg'. Here, f = x^2 and g = y^3. (d)/(dx)(x^2) = 2x. (d)/(dx)(y^3) = 3y^2 (dy)/(dx) (by chain rule). So, (du)/(dx) = (2x)(y^3) + (x^2)(3y^2 (dy)/(dx)) = 2xy^3 + 3x^2 y^2 (dy)/(dx). Therefore, (d)/(dx)((x^2 y^3)) = (x^2 y^3) (2xy^3 + 3x^2 y^2 (dy)/(dx)). Step 3: Differentiate 3x^3 y^2 using the product rule. Here, f = 3x^3 and g = y^2. (d)/(dx)(3x^3) = 9x^2. (d)/(dx)(y^2) = 2y (dy)/(dx) (by chain rule). So, (d)/(dx)(3x^3 y^2) = (9x^2)(y^2) + (3x^3)(2y (dy)/(dx)) = 9x^2 y^2 + 6x^3 y (dy)/(dx). Step 4: Differentiate the constant term. (d)/(dx)(6) = 0. Step 5: Substitute these derivatives back into the equation from Step 1. (x^2 y^3) (2xy^3 + 3x^2 y^2 (dy)/(dx)) + 9x^2 y^2 + 6x^3 y (dy)/(dx) = 0 Step 6: Expand and rearrange the equation to solve for (dy)/(dx). 2xy^3 (x^2 y^3) + 3x^2 y^2 (x^2 y^3) (dy)/(dx) + 9x^2 y^2 + 6x^3 y (dy)/(dx) = 0 Group terms containing (dy)/(dx): (3x^2 y^2 (x^2 y^3) + 6x^3 y) (dy)/(dx) = -2xy^3 (x^2 y^3) - 9x^2 y^2 Factor out (dy)/(dx): (dy)/(dx) (3x^2 y^2 (x^2 y^3) + 6x^3 y) = -(2xy^3 (x^2 y^3) + 9x^2 y^2) Isolate (dy)/(dx): (dy)/(dx) = -(2xy^3 (x^2 y^3) + 9x^