![[x2 + y2]2 = 4x2y](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1780662251841-0ab0968b1ce45147.png&w=3840&q=75)
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.
ScanSolve AI Answer
To find (dy)/(dx) from the given equation, we will use implicit differentiation. Step 1: Write the given equation. (x^2 + y^2)^2 = 4x^2y Step 2: Differentiate both sides of the equation with respect to x. Apply the chain rule to the left side and the product rule to the right side. (d)/(dx)(x^2 + y^2)^2 = (d)/(dx)(4x^2y) 2(x^2 + y^2) · (d)/(dx)(x^2 + y^2) = 4 ( (d)/(dx)(x^2) · y + x^2 · (d)/(dx)(y) ) 2(x^2 + y^2)(2x + 2y (dy)/(dx)) = 4(2xy + x^2 (dy)/(dx)) Step 3: Expand and rearrange the terms to group (dy)/(dx). 4(x^2 + y^2)(x + y (dy)/(dx)) = 8xy + 4x^2 (dy)/(dx) 4x(x^2 + y^2) + 4y(x^2 + y^2) (dy)/(dx) = 8xy + 4x^2 (dy)/(dx) 4x^3 + 4xy^2 + (4x^2y + 4y^3) (dy)/(dx) = 8xy + 4x^2 (dy)/(dx) Move all terms containing (dy)/(dx) to one side and other terms to the other side: (4x^2y + 4y^3) (dy)/(dx) - 4x^2 (dy)/(dx) = 8xy - 4x^3 - 4xy^2 Step 4: Factor out (dy)/(dx) and solve for it. (dy)/(dx)(4x^2y + 4y^3 - 4x^2) = 8xy - 4x^3 - 4xy^2 (dy)/(dx) = (8xy - 4x^3 - 4xy^2)/(4x^2y + 4y^3 - 4x^2) Divide the numerator and denominator by 4 to simplify: (dy)/(dx) = (2xy - x^3 - xy^2)/(x^2y + y^3 - x^2) The derivative (dy)/(dx) is: (dy)/(dx) = (2xy - x^3 - xy^2)/(x^2y + y^3 - x^2) Send me the next one 📸