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
You're on the right track! Let's complete the first part and then move to the next. i) y = x^3 Step 2: Calculate the difference f(x+h) - f(x). f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3) - x^3 = 3x^2h + 3xh^2 + h^3 Step 3: Substitute into the limit definition and simplify. f'(x) = _h 0 (3x^2h + 3xh^2 + h^3)/(h) f'(x) = _h 0 (h(3x^2 + 3xh + h^2))/(h) f'(x) = _h 0 (3x^2 + 3xh + h^2) Step 4: Evaluate the limit. f'(x) = 3x^2 + 3x(0) + (0)^2 f'(x) = 3x^2 ii) y = 3x + 2 Step 1: Identify f(x) and f(x+h). f(x) = 3x + 2 f(x+h) = 3(x+h) + 2 = 3x + 3h + 2 Step 2: Calculate the difference f(x+h) - f(x). f(x+h) - f(x) = (3x + 3h + 2) - (3x + 2) = 3h Step 3: Substitute into the limit definition and simplify. f'(x) = _h 0 (3h)/(h) f'(x) = _h 0 3 Step 4: Evaluate the limit. f'(x) = 3 That's 2 down. 3 left today — send the next one.