Login or Sign Up

Ask a New Question

Find dy/dx if x^2+4xy-y^2=8

pls provide step by step and also use rules! (like product rule)

find dy/dx (implicit differentiation)

Did you know?

Did you know that finding the derivative of a function can be done using various rules, such as the product rule? Let's apply the rules to find dy/dx for the equation x^2 + 4xy - y^2 = 8. Step by step: 1. Start by differentiating both sides of the equation with respect to x. 2. Using the product rule for the term 4xy, consider it as the product of two functions, f(x) = 4x and g(x) = y. 3. Apply the product rule: the derivative of f(x)g(x) is given by f'(x)g(x) + f(x)g'(x). 4. Differentiating f(x) = 4x, we get f'(x) = 4. 5. Differentiating g(x) = y, we get g'(x) = dy/dx. 6. Plugging these values into the product rule formula, we have 4xy + 4(dy/dx)x + 4x(dy/dx) - 2yy' = 0. 7. Rearrange the equation to isolate dy/dx terms: 4x(dy/dx) + 4y + 4xy - 2yy' = 0. 8. Group the terms containing dy/dx: (4x - 2y)dy/dx = -4xy - 4y. 9. Divide both sides by (4x - 2y) to solve for dy/dx: dy/dx = (-4xy - 4y)/(4x - 2y). 10. Simplifying the equation gives: dy/dx = (-2xy - 2y)/(2x - y). So, by applying the product rule and simplifying, we find that the derivative dy/dx for the given equation x^2 + 4xy - y^2 = 8 is (-2xy - 2y)/(2x - y).