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 are the derivatives of the given functions using first principles: 1) f(x) = x Step 1: State the definition of the derivative from first principles. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x) = x and f(x+h) = (x+h) into the formula. f'(x) = _h 0 ((x+h) - x)/(h) Step 3: Apply the sum-to-product trigonometric identity A - B = -2 ((A+B)/(2)) ((A-B)/(2)). f'(x) = _h 0 (-2 (x+h+x)/(2)) ((x+h-x)/(2))h f'(x) = _h 0 (-2 (x+h)/(2)) ((h)/(2))h Step 4: Rearrange the expression to use the fundamental limit _ 0 ( )/() = 1. f'(x) = _h