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
Step 1: Write down the definition of the derivative from first principles. The derivative f'(x) of a function f(x) is given by: f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Identify f(x) and f(x+h). Given f(x) = (7)/(x^2 + 3). Substitute x+h into f(x) to find f(x+h): f(x+h) = (7)/((x+h)^2 + 3) = (7)/(x^2 + 2xh + h^2 + 3) Step 3: Calculate f(x+h) - f(x). f(x+h) - f(x) = (7)/(x^2 + 2xh + h^2 + 3) - (7)/(x^2 + 3) To subtract the fractions, find a common denominator: = (7(x^2 + 3) - 7(x^2 + 2xh + h^2 + 3))/((x^2 + 2xh + h^2 + 3)(x^2 + 3)) Expand the numerator: = (7x^2 + 21 - (7x^2 + 14xh + 7h^2 + 21))/((x^2 + 2xh + h^2 + 3)(x^2 + 3)) = (7x^2 + 21 - 7x^2 - 14xh - 7h^2 - 21)/((x^2 + 2xh + h^2 + 3)(x^2 + 3)) Simplify the numerator: = (-14xh - 7h^2)/((x^2 + 2xh + h^2 + 3)(x^2 + 3)) Factor out h from the numerator: = (h(-14x - 7h))/((x^2 + 2xh + h^2 + 3)(x^2 + 3)) Step 4: Divide by h. (f(x+h) - f(x))/(h) = (h(-14x - 7h))/((x^2 + 2xh + h^2 + 3)(x^2 + 3))h Cancel out h: = (-14x - 7h)/((x^2 + 2xh + h^2 + 3)(x^2 + 3)) Step 5: Take the limit as h 0. f'(x) = _h 0 (-14x - 7h)/((x^2 + 2xh + h^2 + 3)(x^2 + 3)) Substitute h=0 into the expression: f'(x) = (-14x - 7(0))/((x^2 + 2x(0) + (0)^2 + 3)(x^2 + 3)) f'(x) = (-14x)/((x^2 + 3)(x^2 + 3)) f'(x) = (-14x)/((x^2 + 3)^2) The derivative of f(x) = (7)/(x^2 + 3) from first principles is: (-14x)/((x^2 + 3)^2)