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 for question 2.a, differentiating by the first principle: The first principle of differentiation states that for a function Y = f(x), its derivative (dY)/(dx) is given by: (dY)/(dx) = _h 0 (f(x+h) - f(x))/(h) Step 1: Identify f(x) and f(x+h). Given Y = f(x) = 2x^2. Then, f(x+h) = 2(x+h)^2. Step 2: Expand f(x+h). f(x+h) = 2(x^2 + 2xh + h^2) f(x+h) = 2x^2 + 4xh + 2h^2 Step 3: Calculate f(x+h) - f(x). f(x+h) - f(x) = (2x^2 + 4xh + 2h^2) - (2x^2) f(x+h) - f(x) = 4xh + 2h^2 Step 4: Divide by h. (f(x+h) - f(x))/(h) = (4xh + 2h^2)/(h) (f(x+h) - f(x))/(h) = (h(4x + 2h))/(h) (f(x+h) - f(x))/(h) = 4x + 2h Step 5: Take the limit as h 0. (dY)/(dx) = _h 0 (4x + 2h) (dY)/(dx) = 4x + 2(0) (dY)/(dx) = 4x The derivative by first principle is: 4x Drop the next question!