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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Here are the differentiations for each function using implicit differentiation: 41. Differentiate x^2 y + y^3 = 1 Step 1: Differentiate both sides with respect to x. (d)/(dx)(x^2 y) + (d)/(dx)(y^3) = (d)/(dx)(1) Step 2: Apply the product rule for x^2 y and the chain rule for y^3. (2x · y + x^2 · (dy)/(dx)) + 3y^2 (dy)/(dx) = 0 Step 3: Group terms containing (dy)/(dx). x^2 (dy)/(dx) + 3y^2 (dy)/(dx) = -2xy Step 4: Factor out (dy)/(dx). (dy)/(dx)(x^2 + 3y^2) = -2xy Step 5: Solve for (dy)/(dx). (dy)/(dx) = (-2xy)/(x^2 + 3y^2) The derivative is (-2xy)/(x^2 + 3y^2). 42. Differentiate y + x^2 = 3 Step 1: Differentiate both sides with respect to x. (d)/(dx)( y) + (d)/(dx)(x^2) = (d)/(dx)(3) Step 2: Apply the chain rule for y. (1)/(y) (dy)/(dx) + 2x = 0 Step 3: Isolate the term with (dy)/(dx). (1)/(y) (dy)/(dx) = -2x Step 4: Solve for (dy)/(dx). (dy)/(dx) = -2xy The derivative is -2xy. 43. Differentiate e^xy = x + y Step 1: Differentiate both sides with respect to x. (d)/(dx)(e^xy) = (d)/(dx)(x) + (d)/(dx)(y) Step 2: Apply the chain rule and product rule for e^xy. e^xy · (d)/(dx)(xy) = 1 + (dy)/(dx) e^xy (1 · y + x · (dy)/(dx)) = 1 + (dy)/(dx) ye^xy + xe^xy (dy)/(dx) = 1 + (dy)/(dx) Step 3: Group terms containing (dy)/(dx). xe^xy (dy)/(dx) - (dy)/(dx) = 1 - ye^xy Step 4: Factor out (dy)/(dx). (dy)/(dx)(xe^xy - 1) = 1 - ye^xy Step 5: Solve for (dy)/(dx). (dy)/(dx) = 1 - ye^xyxe^xy - 1 The derivative is 1 - ye^xyxe^xy - 1. 44. Differentiate x^2 + y = y Step 1: Differentiate both sides with respect to x. (d)/(dx)(x^2) + (d)/(dx)( y) = (d)/(dx)(y) Step 2: Apply the chain rule for y. 2x + y (dy)/(dx) = (dy)/(dx) Step 3: Group terms containing (dy)/(dx). 2x = (dy)/(dx) - y (dy)/(dx) Step 4: Factor out (dy)/(dx). 2x = (dy)/(dx)(1 - y) Step 5: Solve for (dy)/(dx). (dy)/(dx) = (2x)/(1 - y) The derivative is (2x)/(1 - y). 45. Differentiate x^3 + y^3 + 3xy = 1 Step 1: Differentiate both sides with respect to x. (d)/(dx)(x^3) + (d)/(dx)(y^3) + (d)/(dx)(3xy) = (d)/(dx)(1) Step 2: Apply the chain rule for y^3 and the product rule for 3xy. 3x^2 + 3y^2 (dy)/(dx) + 3(1 · y + x · (dy)/(dx)) = 0 3x^2 + 3y^2 (dy)/(dx) + 3y + 3x (dy)/(dx) = 0 Step 3: Group terms containing (dy)/(dx). 3y^2 (dy)/(dx) + 3x (dy)/(dx) = -3x^2 - 3y Step 4: Factor out (dy)/(dx). (dy)/(dx)(3y^2 + 3x) = -3x^2 - 3y Step 5: Solve for (dy)/(dx). (dy)/(dx) = (-3x^2 - 3y)/(3y^2 + 3x) Step 6: Simplify by dividing the numerator and denominator by 3. (dy)/(dx) = (-(x^2 + y))/(y^2 + x) The derivative is (-(x^2 + y))/(y^2 + x). That's 2 down. 3 left today — send the next one.