Differentiate the equation (x2 y3) + 3x3 y2 = 6 with respect to x and solve for (dy)/(dx).

Step 1: Differentiate both sides of the equation with respect to x. Given the equation: (x^2 y^3) + 3x^3 y^2 = 6 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 the first term, (x^2 y^3), using the chain rule. Let u = x^2 y^3. Then (d)/(dx)( u) = u · (du)/(dx). First, find (du)/(dx) = (d)/(dx)(x^2 y^3) using the product rule, where y is a function of x. (d)/(dx)(x^2 y^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^2 y^2 (dy)/(dx) Now substitute this back into the derivative of the first term: (d)/(dx)((x^2 y^3)) = (x^2 y^3) (2xy^3 + 3x^2 y^2 (dy)/(dx)) = 2xy^3 (x^2 y^3) + 3x^2 y^2 (x^2 y^3) (dy)/(dx) Step 3: Differentiate the second term, 3x^3 y^2, using the product rule. (d)/(dx)(3x^3 y^2) = ((d)/(dx) 3x^3) y^2 + 3x^3 ((d)/(dx) 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: Combine all differentiated terms and solve for (dy)/(dx). Substitute the results from Step 2, Step 3, and Step 4 back into the equation from Step 1: 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) on one side and other terms on the other side: (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) = (-2xy^3 (x^2 y^3) - 9x^2 y^2)/(3x^2 y^2 (x^2 y^3) + 6x^3 y) Step 6: Simplify the expression. Factor out common terms from the numerator and the denominator. Numerator: -xy^2 (2y (x^2 y^3) + 9x) Denominator: 3x^2 y (y (x^2 y^3) + 2x) (dy)/(dx) = (-xy^2 (2y (x^2 y^3) + 9x))/(3x^2 y (y (x^2 y^3) + 2x)) Cancel out common factors (xy^2)/(x^2 y) = (y)/(x): (dy)/(dx) = (-y (2y (x^2 y^3) + 9x))/(3x (y (x^2 y^3) + 2x)) The final answer is (dy)/(dx) = (-y (2y (x^2 y^3) + 9x))/(3x (y (x^2 y^3) + 2x)). Send me the next one 📸