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.
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To find the derivative of f(x) = x^2 + (3)/(x) with respect to x from first principles, we use the definition: f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 1: Write down f(x) and f(x+h). Given f(x) = x^2 + (3)/(x). Then f(x+h) = (x+h)^2 + (3)/(x+h). Expand (x+h)^2: f(x+h) = x^2 + 2xh + h^2 + (3)/(x+h) Step 2: Calculate f(x+h) - f(x). f(x+h) - f(x) = (x^2 + 2xh + h^2 + (3)/(x+h)) - (x^2 + (3)/(x)) f(x+h) - f(x) = x^2 + 2xh + h^2 + (3)/(x+h) - x^2 - (3)/(x) f(x+h) - f(x) = 2xh + h^2 + (3)/(x+h) - (3)/(x) Combine the fractional terms: f(x+h) - f(x) = 2xh + h^2 + (3x - 3(x+h))/(x(x+h)) f(x+h) - f(x) = 2xh + h^2 + (3x - 3x - 3h)/(x(x+h)) f(x+h) - f(x) = 2xh + h^2 - (3h)/(x(x+h)) Step 3: Divide by h. (f(x+h) - f(x))/(h) = (2xh + h^2 - 3h)/(x(x+h))h Factor out h from the numerator: (f(x+h) - f(x))/(h) = (h(2x + h) - 3h)/(x(x+h))h (f(x+h) - f(x))/(h) = 2x + h - (3)/(x(x+h)) Step 4: Take the limit as h 0. f'(x) = _h 0 (2x + h - (3)/(x(x+h))) Substitute h=0 into the expression: f'(x) = 2x + 0 - (3)/(x(x+0)) f'(x) = 2x - (3)/(x^2) The derivative of x^2 + (3)/(x) from first principles is 2x - (3)/(x^2). 3 done, 2 left today. You're making progress.