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 derivations using the first principle: 1. For F(x) = x^2 - x: The first principle definition is F'(x) = _h 0 (F(x+h) - F(x))/(h). Step 1: Find F(x+h). F(x+h) = (x+h)^2 - (x+h) = x^2 + 2xh + h^2 - x - h Step 2: Find F(x+h) - F(x). F(x+h) - F(x) = (x^2 + 2xh + h^2 - x - h) - (x^2 - x) = 2xh + h^2 - h Step 3: Divide by h. (F(x+h) - F(x))/(h) = (2xh + h^2 - h)/(h) = 2x + h - 1 Step 4: Take the limit as h 0. F'(x) = _h 0 (2x + h - 1) = 2x - 1 The derivative is 2x-1. 2. For F(x) = sqrt(x+1): The first principle definition is F'(x) = _h 0 (F(x+h) - F(x))/(h). Step 1: Find F(x+h) - F(x). F(x+h) - F(x) = sqrt((x+h)+1) - sqrt(x+1) Step 2: Multiply the numerator and denominator by the conjugate of the numerator. sqrt(x+h+1) - sqrt(x+1)h × sqrt(x+h+1) + sqrt(x+1)sqrt(x+h+1) + sqrt(x+1) Step 3: Simplify the numerator using (a-b)(a+b) = a^2-b^2. ((x+h+1) - (x+1))/(h(sqrt(x+h+1) + x+1)) = (h)/(h(sqrt(x+h+1) + x+1)) Step 4: Cancel h from the numerator and denominator. (1)/(sqrt(x+h+1) + x+1) Step 5: Take the limit as h 0. F'(x) = _h 0 (1)/(sqrt(x+h+1) + x+1) = (1)/(sqrt(x+0+1) + x+1) = (1)/(2sqrt(x+1)) The derivative is (1)/(2sqrt(x+1)). 3. For g(x) = (1)/(x^2): The first principle definition is g'(x) = _h 0 (g(x+h) - g(x))/(h). Step 1: Find g(x+h) - g(x). g(x+h) - g(x) = (1)/((x+h)^2) - (1)/(x^2) Step 2: Combine the fractions in the numerator. (x^2 - (x+h)^2)/(x^2(x+h)^2) = (x^2 - (x^2 + 2xh + h^2))/(x^2(x+h)^2) = (-2xh - h^2)/(x^2(x+h)^2) Step 3: Divide by h. (g(x+h) - g(x))/(h) = (-2xh - h^2)/(h · x^2(x+h)^2) = (h(-2x - h))/(h · x^2(x+h)^2) = (-2x - h)/(x^2(x+h)^2) Step 4: Take the limit as h 0. g'(x) = _h 0 (-2x - h)/(x^2(x+h)^2) = (-2x - 0)/(x^2(x+0)^2) = (-2x)/(x^2 · x^2) = (-2x)/(x^4) = -(2)/(x^3) The derivative is -(2)/(x^3). 4. For h(x) = (1)/(sqrt(x)): The first principle definition is h'(x) = _h 0 (h(x+h) - h(x))/(h). Step 1: Find h(x+h) - h(x). h(x+h) - h(x) = (1)/(sqrt(x+h)) - (1)/(sqrt(x)) Step 2: Combine the fractions in the numerator. sqrt(x) - sqrt(x+h)sqrt(x)sqrt(x+h) Step 3: Multiply the numerator and denominator by the conjugate of the numerator. sqrt(x) - sqrt(x+h)sqrt(x)sqrt(x+h) × sqrt(x) + sqrt(x+h)sqrt(x) + sqrt(x+h) = (x - (x+h))/(sqrt(x)x+h)(sqrt(x) + sqrt(x+h)) = (-h)/(sqrt(x)x+h)(sqrt(x) + sqrt(x+h)) Step 4: Divide by h. (h(x+h) - h(x))/(h) = (-h)/(h · sqrt(x)x+h)(sqrt(x) + sqrt(x+h)) = (-1)/(sqrt(x)x+h)(sqrt(x) + sqrt(x+h)) Step 5: Take the limit as h 0. h'(x) = _h 0 (-1)/(sqrt(x)x+h)(sqrt(x) + sqrt(x+h)) = (-1)/(sqrt(x)x+0)(sqrt(x) + sqrt(x+0)) = (-1)/(sqrt(x)x)(2sqrt(x)) = (-1)/(x(2sqrt(x))) = -(1)/(2x^3/2) The derivative is -(1)/(2x^3/2). 5. For y = (1)/(2x-1): Let f(x) = (1)/(2x-1). The first principle definition is f'(x) = _h 0 (f(x+h) - f(x))/(h). Step 1: Find f(x+h) - f(x). f(x+h) - f(x) = (1)/(2(x+h)-1) - (1)/(2x-1) = (1)/(2x+2h-1) - (1)/(2x-1) Step 2: Combine the fractions in the numerator. ((2x-1) - (2x+2h-1))/((2x+2h-1)(2x-1)) = (2x-1-2x-2h+1)/((2x+2h-1)(2x-1)) = (-2h)/((2x+2h-1)(2x-1)) Step 3: Divide by h. (f(x+h) - f(x))/(h) = (-2h)/(h(2x+2h-1)(2x-1)) = (-2)/((2x+2h-1)(2x-1)) Step 4: Take the limit as h 0. f'(x) = _h 0 (-2)/((2x+2h-1)(2x-1)) = (-2)/((2x+2(0)-1)(2x-1)) = (-2)/((2x-1)(2x-1)) = (-2)/((2x-1)^2) The derivative is -(2)/((2x-1)^2). What's next?