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 the derivative of y = 2x^2 - x + 3 from first principles, we use the definition: f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 1: Identify f(x) and f(x+h). Given f(x) = 2x^2 - x + 3. Substitute (x+h) into f(x) to find f(x+h): f(x+h) = 2(x+h)^2 - (x+h) + 3 f(x+h) = 2(x^2 + 2xh + h^2) - x - h + 3 f(x+h) = 2x^2 + 4xh + 2h^2 - x - h + 3 Step 2: Calculate f(x+h) - f(x). f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - x - h + 3) - (2x^2 - x + 3) f(x+h) - f(x) = 2x^2 + 4xh + 2h^2 - x - h + 3 - 2x^2 + x - 3 f(x+h) - f(x) = 4xh + 2h^2 - h Step 3: Substitute into the first principles formula. f'(x) = _h 0 (4xh + 2h^2 - h)/(h) Step 4: Factor out h from the numerator and simplify. f'(x) = _h 0 (h(4x + 2h - 1))/(h) Since h 0 but h ≠ 0, we can cancel h: f'(x) = _h 0 (4x + 2h - 1) Step 5: Evaluate the limit by substituting h=0. f'(x) = 4x + 2(0) - 1 f'(x) = 4x - 1 The derivative of y = 2x^2 - x + 3 is 4x - 1. 3 done, 2 left today. You're making progress.