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's how to find the derivative of F(x) = x^2 - x using the first principle. Step 1: State the definition of the derivative using the first principle. The derivative F'(x) is defined as: F'(x) = _h 0 (F(x+h) - F(x))/(h) Step 2: Substitute F(x+h) and F(x) into the definition and simplify the numerator. Given F(x) = x^2 - x, we find F(x+h) = (x+h)^2 - (x+h). F'(x) = _h 0 (((x+h)^2 - (x+h)) - (x^2 - x))/(h) Expand (x+h)^2: F'(x) = _h 0 ((x^2 + 2xh + h^2 - x - h) - (x^2 - x))/(h) Distribute the negative sign and combine like terms in the numerator: F'(x) = _h 0 (x^2 + 2xh + h^2 - x - h - x^2 + x)/(h) F'(x) = _h 0 (2xh + h^2 - h)/(h) Step 3: Factor out h from the numerator and cancel it with the denominator. F'(x) = _h 0 (h(2x + h - 1))/(h) Since h 0 but h ≠ 0, we can cancel h: F'(x) = _h 0 (2x + h - 1) Step 4: Evaluate the limit by substituting h=0. F'(x) = 2x + 0 - 1 F'(x) = 2x - 1 That's 2 down. 3 left today — send the next one.