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
Hey bjosiah221, good to see you again. To find the derivative of y = (1)/(x) from first principles, we use the definition: f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 1: Identify f(x) and f(x+h). Given f(x) = (1)/(x), then f(x+h) = (1)/(x+h). Step 2: Substitute f(x) and f(x+h) into the first principles formula. f'(x) = _h 0 (1)/(x+h) - (1)/(x)h Step 3: Simplify the numerator by finding a common denominator. f'(x) = _h 0 (x - (x+h))/(x(x+h))h f'(x) = _h 0 (x - x - h)/(x(x+h))h f'(x) = _h 0 (-h)/(x(x+h))h Step 4: Simplify the complex fraction. f'(x) = _h 0 (-h)/(x(x+h)) · (1)/(h) f'(x) = _h 0 (-1)/(x(x+h)) Step 5: Evaluate the limit as h 0. Substitute h=0 into the expression. f'(x) = (-1)/(x(x+0)) f'(x) = (-1)/(x(x)) f'(x) = -(1)/(x^2) Send me the next one 📸