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
You're on a roll — To find (dy)/(dx) for the given equation, we will use implicit differentiation, applying the chain rule and product rule where necessary. Given equation: (x^2y^3) + 3x^3y^2 = 6 Step 1: Differentiate both sides of the equation with respect to x. (d)/(dx) ( (x^2y^3) + 3x^3y^2 ) = (d)/(dx) (6) Step 2: Differentiate the first term, (x^2y^3), using 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: (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^2y^2 (dy)/(dx). Therefore, the derivative of the first term is: (d)/(dx)((x^2y^3)) = (x^2y^3) (2xy^3 + 3x^2y^2 (dy)/(dx)) = 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) Step 3: Differentiate the second term, 3x^3y^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, the derivative of the second term is: (d)/(dx)(3x^3y^2) = (9x^2)(y^2) + (3x^3)(2y (dy)/(dx)) = 9x^2y^2 + 6x^3y (dy)/(dx) Step 4: Differentiate the right side of the equation. (d)/(dx)(6) = 0 Step 5: Combine the differentiated terms and set the sum to zero. 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) + 9x^2y^2 + 6x^3y (dy)/(dx) = 0 Step 6: Rearrange the equation to isolate terms containing (dy)/(dx). 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 7: Factor out (dy)/(dx) from the terms on the left side. (dy)/(dx) (3x^2y^2 (x^2y^3) + 6x^3y) = -2xy^3 (x^2y^3) - 9x^2y^2 Step 8: Solve for (dy)/(dx). (dy)/(dx) = (-2xy^3 (x^2y^3) - 9x^2y^2)/(3x^2y^2 (x^2y^3) + 6x^3y) Step 9: Simplify the expression by factoring out 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)) Cancel out xy (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)). Send me the next one 📸