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
Step 1: Define the function. Let f(x) = x^2. Step 2: Write down the formula for the derivative using the first principle. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 3: Find f(x+h). Substitute (x+h) into the function f(x): f(x+h) = (x+h)^2 Expand (x+h)^2: (x+h)^2 = x^2 + 2xh + h^2 Step 4: Substitute f(x+h) and f(x) into the first principle formula. f'(x) = _h 0 ((x^2 + 2xh + h^2) - x^2)/(h) Step 5: Simplify the numerator. f'(x) = _h 0 (2xh + h^2)/(h) Step 6: Factor out h from the numerator. f'(x) = _h 0 (h(2x + h))/(h) Step 7: Cancel out h (since h ≠ 0 as h 0). f'(x) = _h 0 (2x + h) Step 8: Evaluate the limit by substituting h=0. f'(x) = 2x + 0 f'(x) = 2x The derivative of x^2 using the first principle is: 2x