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
Step 1: Differentiate both sides of the equation 2x + 3xy = 2y^2 with respect to x. We will apply the differentiation rules: • (d)/(dx)(c) = 0 • (d)/(dx)(cx) = c • (d)/(dx)(cy) = c(dy)/(dx) • (d)/(dx)(u · v) = u'v + uv' (product rule) • (d)/(dx)(y^n) = ny^n-1(dy)/(dx) (chain rule) Differentiating each term: • For 2x: (d)/(dx)(2x) = 2 • For 3xy: Use the product rule with u=3x and v=y. u' = (d)/(dx)(3x) = 3 v' = (d)/(dx)(y) = (dy)/(dx) So, (d)/(dx)(3xy) = (3)(y) + (3x)((dy)/(dx)) = 3y + 3x(dy)/(dx) • For 2y^2: Use the chain rule. (d)/(dx)(2y^2) = 2 · 2y^2-1 · (dy)/(dx) = 4y(dy)/(dx) Step 2: Substitute the differentiated terms back into the equation. 2 + 3y + 3x(dy)/(dx) = 4y(dy)/(dx) Step 3: Rearrange the equation to group terms containing (dy)/(dx) on one side and other terms on the other side. Subtract 3x(dy)/(dx) from both sides: 2 + 3y = 4y(dy)/(dx) - 3x(dy)/(dx) Step 4: Factor out (dy)/(dx) from the terms on the right side. 2 + 3y = (dy)/(dx)(4y - 3x) Step 5: Solve for (dy)/(dx) by dividing both sides by (4y - 3x). (dy)/(dx) = (2 + 3y)/(4y - 3x) The derivative (dy)/(dx) is: (2 + 3y)/(4y - 3x) 3 done, 2 left today. You're making progress.