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 find the derivative of f(x) = (1)/(x^2) using the first principle, 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^2). Then f(x+h) = (1)/((x+h)^2). Step 2: Substitute f(x) and f(x+h) into the first principle formula. f'(x) = _h 0 (1)/((x+h)^2) - (1)/(x^2)h Step 3: Simplify the numerator by finding a common denominator. f'(x) = _h 0 (x^2 - (x+h)^2)/(x^2(x+h)^2)h f'(x) = _h 0 (x^2 - (x^2 + 2xh + h^2))/(h x^2(x+h)^2) f'(x) = _h 0 (x^2 - x^2 - 2xh - h^2)/(h x^2(x+h)^2) f'(x) = _h 0 (-2xh - h^2)/(h x^2(x+h)^2) Step 4: Factor out h from the numerator and cancel it with the h in the denominator. f'(x) = _h 0 (h(-2x - h))/(h x^2(x+h)^2) f'(x) = _h 0 (-2x - h)/(x^2(x+h)^2) Step 5: Evaluate the limit by substituting h=0. f'(x) = (-2x - 0)/(x^2(x+0)^2) f'(x) = (-2x)/(x^2(x^2)) f'(x) = (-2x)/(x^4) f'(x) = -(2)/(x^3) The derivative of (1)/(x^2) using the first principle is: -(2)/(x^3) Send me the next one 📸