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 y^2 + 3x + 2xy = 5x^2 with respect to x. We will apply the differentiation rules: • (d)/(dx)(y^n) = ny^n-1(dy)/(dx) (chain rule) • (d)/(dx)(cx) = c • (d)/(dx)(u · v) = u'v + uv' (product rule) • (d)/(dx)(cx^n) = ncx^n-1 Differentiating each term: • For y^2: (d)/(dx)(y^2) = 2y(dy)/(dx) • For 3x: (d)/(dx)(3x) = 3 • For 2xy: Use the product rule with u=2x and v=y. u' = (d)/(dx)(2x) = 2 v' = (d)/(dx)(y) = (dy)/(dx) So, (d)/(dx)(2xy) = (2)(y) + (2x)((dy)/(dx)) = 2y + 2x(dy)/(dx) • For 5x^2: (d)/(dx)(5x^2) = 5 · 2x^2-1 = 10x Step 2: Substitute the differentiated terms back into the equation. 2y(dy)/(dx) + 3 + 2y + 2x(dy)/(dx) = 10x Step 3: Rearrange the equation to group terms containing (dy)/(dx) on one side and other terms on the other side. Subtract 3 and 2y from both sides: 2y(dy)/(dx) + 2x(dy)/(dx) = 10x - 3 - 2y Step 4: Factor out (dy)/(dx) from the terms on the left side. (dy)/(dx)(2y + 2x) = 10x - 3 - 2y Step 5: Solve for (dy)/(dx) by dividing both sides by (2y + 2x). (dy)/(dx) = (10x - 3 - 2y)/(2y + 2x) The derivative (dy)/(dx) is: (10x - 3 - 2y)/(2x + 2y) What's next?