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 with respect to x: 1) For the equation x^2 + y^2 = x + 4 + 8: Step 1: Simplify the equation. x^2 + y^2 = x + 12 Step 2: Differentiate both sides with respect to x. Remember to use the chain rule for terms involving y. (d)/(dx)(x^2) + (d)/(dx)(y^2) = (d)/(dx)(x) + (d)/(dx)(12) 2x + 2y (dy)/(dx) = 1 + 0 Step 3: Isolate (dy)/(dx). 2y (dy)/(dx) = 1 - 2x (dy)/(dx) = (1 - 2x)/(2y) The derivative is: (dy)/(dx) = (1 - 2x)/(2y) 2) For the equation x^3 + y^3 = x^3 y^3 - 1: Step 1: Differentiate both sides with respect to x. Use the chain rule for y terms and the product rule for x^3 y^3. (d)/(dx)(x^3) + (d)/(dx)(y^3) = (d)/(dx)(x^3 y^3) - (d)/(dx)(1) 3x^2 + 3y^2 (dy)/(dx) = (3x^2 y^3 + x^3 · 3y^2 (dy)/(dx)) - 0 3x^2 + 3y^2 (dy)/(dx) = 3x^2 y^3 + 3x^3 y^2 (dy)/(dx) Step 2: Group terms containing (dy)/(dx) on one side and other terms on the other side. 3y^2 (dy)/(dx) - 3x^3 y^2 (dy)/(dx) = 3x^2 y^3 - 3x^2 Step 3: Factor out (dy)/(dx) and solve for it. (dy)/(dx) (3y^2 - 3x^3 y^2) = 3x^2 y^3 - 3x^2 (dy)/(dx) = (3x^2 y^3 - 3x^2)/(3y^2 - 3x^3 y^2) Step 4: Simplify the expression by factoring out common terms. (dy)/(dx) = (3x^2 (y^3 - 1))/(3y^2 (1 - x^3)) (dy)/(dx) = (x^2 (y^3 - 1))/(y^2 (1 - x^3)) The derivative is: (dy)/(dx) = (x^2 (y^3 - 1))/(y^2 (1 - x^3)) 3) For the equation x^y = y^x: Step 1: Take the natural logarithm of both sides to bring down the exponents. (x^y) = (y^x) y x = x y Step 2: Differentiate both sides with respect to x using the product rule. (d)/(dx)(y x) = (d)/(dx)(x y) ((dy)/(dx) · x + y · (1)/(x)) = (1 · y + x · (1)/(y) (dy)/(dx)) x (dy)/(dx) + (y)/(x) = y + (x)/(y) (dy)/(dx) Step 3: Group terms containing (dy)/(dx) on one side and other terms on the other side. x (dy)/(dx) - (x)/(y) (dy)/(dx) = y - (y)/(x) Step 4: Factor out (dy)/(dx) and solve for it. (dy)/(dx) ( x - (x)/(y)) = y - (y)/(x) (dy)/(dx) ((y x - x)/(y)) = (x y - y)/(x) (dy)/(dx) = (x y - y)/(x)(y x - x)/(y) (dy)/(dx) = (x y - y)/(x) · (y)/(y x - x) (dy)/(dx) = (y(x y - y))/(x(y x - x)) The derivative is: (dy)/(dx) = (y(x y - y))/(x(y x - x)) That's 2 down. 3 left today — send the next one.