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
Morning Ellison — let's get this done. To differentiate f(x) = 2x^2 - 16 from the first principle, we use the definition: f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 1: Find f(x+h). Substitute (x+h) into the function f(x). f(x+h) = 2(x+h)^2 - 16 Expand the term (x+h)^2: f(x+h) = 2(x^2 + 2xh + h^2) - 16 f(x+h) = 2x^2 + 4xh + 2h^2 - 16 Step 2: Calculate f(x+h) - f(x). Subtract the original function f(x) from f(x+h). f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - 16) - (2x^2 - 16) f(x+h) - f(x) = 2x^2 + 4xh + 2h^2 - 16 - 2x^2 + 16 f(x+h) - f(x) = 4xh + 2h^2 Step 3: Divide by h. Divide the result from Step 2 by h. (f(x+h) - f(x))/(h) = (4xh + 2h^2)/(h) Factor out h from the numerator: (f(x+h) - f(x))/(h) = (h(4x + 2h))/(h) Cancel h (since h ≠ 0 in the limit process): (f(x+h) - f(x))/(h) = 4x + 2h Step 4: Take the limit as h 0. Apply the limit to the expression from Step 3. f'(x) = _h 0 (4x + 2h) Substitute h=0 into the expression: f'(x) = 4x + 2(0) f'(x) = 4x The derivative of f(x) = 2x^2 - 16 from the first principle is 4x. What's next?