answer:To find the value of y when x = 3 and dy/dx = -2/3, we first need to find the derivative of the given implicit function with respect to x. Given equation: x^3 + y^3 = 54 Differentiate both sides with respect to x: d/dx(x^3) + d/dx(y^3) = d/dx(54) Using the chain rule for the derivative of y^3 with respect to x, we get: 3x^2 + 3y^2(dy/dx) = 0 Now, we are given that dy/dx = -2/3 when x = 3. We can plug these values into the equation: 3(3)^2 + 3y^2(-2/3) = 0 27 - 2y^2 = 0 2y^2 = 27 y^2 = 27/2 y = ±√(27/2) Now, we need to determine which value of y satisfies the original equation when x = 3: (3)^3 + y^3 = 54 27 + y^3 = 54 y^3 = 27 y = 3 So, the value of y is 3 when x = 3 and dy/dx = -2/3.

answer:To find frac{dy}{dx}, we will differentiate the given implicit function with respect to x. We will use the chain rule and the product rule when necessary. Given function: xy^3 + 2x^2y - 3x = 7 Differentiating with respect to x: frac{d}{dx}(xy^3) + frac{d}{dx}(2x^2y) - frac{d}{dx}(3x) = frac{d}{dx}(7) Using the product rule for the first term: (1)(y^3) + x(3y^2)frac{dy}{dx} Using the product rule for the second term: (4x)(y) + 2x^2frac{dy}{dx} Differentiating the third term: -3 Differentiating the constant term: 0 Now, combining the terms: y^3 + 3xy^2frac{dy}{dx} + 4xy + 2x^2frac{dy}{dx} - 3 = 0 Now, we need to find frac{dy}{dx} at the point (1, 2). Substitute x = 1 and y = 2: 2^3 + 3(1)(2^2)frac{dy}{dx} + 4(1)(2) + 2(1)^2frac{dy}{dx} - 3 = 0 Simplifying: 8 + 12frac{dy}{dx} + 8 + 2frac{dy}{dx} - 3 = 0 Combine the terms with frac{dy}{dx}: 14frac{dy}{dx} = -13 Now, solve for frac{dy}{dx}: frac{dy}{dx} = -frac{13}{14} So, the derivative frac{dy}{dx} at the point (1, 2) is -frac{13}{14}.

answer:To find the derivative of y with respect to x (dy/dx), we'll first differentiate both sides of the given equation with respect to x using implicit differentiation. The given equation is: x^2 * y - y^2 * sin(x) = 5 Differentiating both sides with respect to x, we get: (d/dx)(x^2 * y) - (d/dx)(y^2 * sin(x)) = (d/dx)(5) Using the product rule for differentiation, we have: [(2x * y) + (x^2 * dy/dx)] - [(2y * dy/dx * sin(x)) + (y^2 * cos(x))] = 0 Now, we need to find dy/dx at the point (pi, 1). So, we'll substitute x = pi and y = 1 into the equation: (2 * pi * 1) + (pi^2 * dy/dx) - (2 * 1 * dy/dx * sin(pi)) - (1^2 * cos(pi)) = 0 Simplifying the equation, we get: (2 * pi) + (pi^2 * dy/dx) - (0) - (-1) = 0 Now, solve for dy/dx: (2 * pi) + 1 = (pi^2 * dy/dx) Divide both sides by pi^2: dy/dx = (2 * pi + 1) / pi^2 So, the derivative of y with respect to x (dy/dx) at the point (pi, 1) is: dy/dx = (2 * pi + 1) / pi^2

answer:To find the derivative of y with respect to x, we'll first differentiate both sides of the equation with respect to x using the chain rule. Given equation: y^3 + xy - 4 = 0 Differentiating both sides with respect to x: d/dx(y^3) + d/dx(xy) - d/dx(4) = d/dx(0) For the first term, we have: d/dx(y^3) = 3y^2 * dy/dx (using the chain rule) For the second term, we have: d/dx(xy) = x(dy/dx) + y (using the product rule) The third term is a constant, so its derivative is 0: d/dx(4) = 0 Now, we can rewrite the differentiated equation: 3y^2 * dy/dx + x(dy/dx) + y = 0 Now, we want to find dy/dx at the point (2, 1). We can plug in x = 2 and y = 1 into the equation: 3(1)^2 * dy/dx + 2(dy/dx) + 1 = 0 Simplifying the equation: 3(dy/dx) + 2(dy/dx) + 1 = 0 Combining the dy/dx terms: 5(dy/dx) = -1 Now, we can solve for dy/dx: dy/dx = -1/5 So, the derivative of y with respect to x at the point (2, 1) is -1/5.