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: Find the partial derivative of f(x,y) with respect to y. The given function is f(x,y) = e^2y (xy). We use the product rule ()/( y)(uv) = ( u)/( y)v + u( v)/( y). Let u = e^2y and v = (xy). First, find ( u)/( y): ()/( y)(e^2y) = e^2y · ()/( y)(2y) = 2e^2y Next, find ( v)/( y): ()/( y)((xy)) = -(xy) · ()/( y)(xy) = -(xy) · x = -x(xy) Now, apply the product rule: f_y(x,y) = (2e^2y)(xy) + e^2y(-x(xy)) f_y(x,y) = 2e^2y(xy) - xe^2y(xy) Factor out e^2y: f_y(x,y) = e^2y(2(xy) - x(xy)) Step 2: Evaluate f_y(x,y) at the point (0, ). Substitute x=0 and y= into the expression for f_y(x,y): f_y(0, ) = e^2()(2(0 · ) - 0 · (0 · )) f_y(0, ) = e^2(2(0) - 0 · (0)) Since (0) = 1 and (0) = 0: f_y(0, ) = e^2(2(1) - 0(0)) f_y(0, ) = e^2(2 - 0) f_y(0, ) = 2e^2 The final answer is 2e^2. Send me the next one 📸