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^2y^3) + 3x^3y^2 = 6, we will use implicit differentiation, applying the chain rule and product rule. 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 each term. For the first term, (d)/(dx)((x^2y^3)), we use the chain rule. Let u = x^2y^3. (d)/(dx)((u)) = (u) · (du)/(dx) Now, we need to find (du)/(dx) = (d)/(dx)(x^2y^3) using the product rule. (d)/(dx)(x^2y^3) = (d)/(dx)(x^2) · y^3 + x^2 · (d)/(dx)(y^3) = (2x)y^3 + x^2(3y^2 (dy)/(dx)) = 2xy^3 + 3x^2y^2 (dy)/(dx) So, 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), we use the product rule. (d)/(dx)(3x^3y^2) = (d)/(dx)(3x^3) · y^2 + 3x^3 · (d)/(dx)(y^2) = (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 and set the equation to zero. (x^2y^3) (2xy^3 + 3x^2y^2 (dy)/(dx)) + 9x^2y^2 + 6x^3y (dy)/(dx) = 0 Distribute (x^2y^3): 2xy^3 (x^2y^3) + 3x^2y^2 (x^2y^3) (dy)/(dx) + 9x^2y^2 + 6x^3y (dy)/(dx) = 0 Step 4: 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): (3x^2y^2 (x^2y^3) + 6x^3y) (dy)/(dx) = -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 out common terms from the numerator and denominator. Factor out -xy^2 from the numerator: -xy^2 (2y (x^2y^3) + 9x) Factor out 3x^2y from the denominator: 3x^2y (y (x^2y^3) + 2x) So, (dy)/(dx) = (-xy^2 (2y (x^2y^3) + 9x))/(3x^2y (y (x^2y^3) + 2x)) Cancel xy from the numerator and denominator (assuming x ≠ 0 and y ≠ 0): (dy)/(dx) = (-y (2y (x^2y^3) + 9x))/(3x (y (x^2y^3) + 2x)) The final answer is (dy)/(dx) = (-y (2y (x^2y^3) + 9x))/(3x (y (x^2y^3) + 2x)). Last free one today — make it count tomorrow, or type /upgrade for unlimited.