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 determine f'(x) from first principles, we use the definition of the derivative: f'(x) = _h 0 (f(x+h) - f(x))/(h) Given f(x) = -2x+3. Step 1: Find f(x+h). Substitute x+h into the function f(x): f(x+h) = -2(x+h) + 3 f(x+h) = -2x - 2h + 3 Step 2: Find the difference f(x+h) - f(x). f(x+h) - f(x) = (-2x - 2h + 3) - (-2x + 3) f(x+h) - f(x) = -2x - 2h + 3 + 2x - 3 f(x+h) - f(x) = -2h Step 3: Form the difference quotient (f(x+h) - f(x))/(h). (f(x+h) - f(x))/(h) = (-2h)/(h) (f(x+h) - f(x))/(h) = -2 Step 4: Take the limit as h 0. f'(x) = _h 0 (-2) Since the expression does not contain h, the limit is the constant value itself. f'(x) = -2 The derivative from first principles is -2.