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
Here's the solution to the problem. Given the function z = x^2 + y^3 + x^2 + y^3. Step 1: Simplify the expression for z. z = 2x^2 + 2y^3 i) Evaluate (dz)/(dx). Step 1: Differentiate z with respect to x, treating y as a constant. (dz)/(dx) = ()/( x)(2x^2 + 2y^3) (dz)/(dx) = ()/( x)(2x^2) + ()/( x)(2y^3) (dz)/(dx) = 4x + 0 (dz)/(dx) = 4x The partial derivative (dz)/(dx) is 4x. ii) Evaluate (dz)/(dy) when x=1 and y=1. Step 1: Differentiate z with respect to y, treating x as a constant. (dz)/(dy) = ()/( y)(2x^2 + 2y^3) (dz)/(dy) = ()/( y)(2x^2) + ()/( y)(2y^3) (dz)/(dy) = 0 + 6y^2 (dz)/(dy) = 6y^2 Step 2: Substitute x=1 and y=1 into the expression for (dz)/(dy). (dz)/(dy) |_(x=1, y=1) = 6(1)^2 (dz)/(dy) |_(x=1, y=1) = 6(1) (dz)/(dy) |_(x=1, y=1) = 6 The partial derivative (dz)/(dy) at x=1, y=1 is 6. That's 2 down. 3 left today — send the next one.