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. We apply the derivative operator (d)/(dx) to both sides: (d)/(dx)((x^2y^3)) + (d)/(dx)(3x^3y^2) = (d)/(dx)(6) Step 2: Differentiate each term. For the first term, (d)/(dx)((x^2y^3)), we use the chain rule. Let u = x^2y^3. Then (d)/(dx)((u)) = (u) (du)/(dx). To find (du)/(dx) = (d)/(dx)(x^2y^3), we use the product rule, where f = x^2 and g = y^3. (df)/(dx) = 2x. (dg)/(dx) = 3y^2 (dy)/(dx) (using the chain rule for y^3). So, (du)/(dx) = (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)). For the second term, (d)/(dx)(3x^3y^2), we use the product rule, where f = 3x^3 and g = y^2. (df)/(dx) = 9x^2. (dg)/(dx) = 2y (dy)/(dx) (using the chain rule for y^2). So, (d)/(dx)(3x^3y^2) = (9x^2)(y^2) + (3x^3)(2y (dy)/(dx)) = 9x^2y^2 + 6x^3y (dy)/(dx). For the right side, (d)/(dx)(6) = 0. Step 3: Substitute the differentiated terms back into the equation. (x^2y^3) (2xy^3 + 3x^2y^2 (dy)/(dx)) + 9x^2y^2 + 6x^3y (dy)/(dx) = 0 Step 4: Expand and rearrange the equation to isolate (dy)/(dx). 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) + 9x^2y^2 + 6x^3y (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^3y (dy)/(dx) = -2xy^3 (x^2y^3) - 9x^2y^2 Factor out (dy)/(dx): (dy)/(dx) (3x^2y^2 (x^2y^3) + 6x^3y) = -2xy^3 (x^2y^3) - 9x^2y^2 Step 5: Solve for (dy)/(dx). (dy)/(dx) = (-2xy^3 (x^2y^3) - 9x^2y^2)/(3x^2y^2 (x^2y^3) + 6x^3y) Step 6: Simplify the expression by factoring common terms from the numerator and denominator. Factor out -xy^2 from the numerator: Numerator = -xy^2 (2y (x^2y^3) + 9x). Factor out 3xy from the denominator: Denominator = 3xy (xy (x^2y^3) + 2x^2). Substitute these back into the expression for (dy)/(dx): (dy)/(dx) = (-xy^2 (2y (x^2y^3) + 9x))/(3xy (xy (x^2y^3) + 2x^2)) Assuming x ≠ 0 and y ≠ 0, we can cancel xy from the numerator and denominator: (dy)/(dx) = -y (2y (x^2y^3) + 9x)3 (xy (