Define the function and the first principles formula.

Step 1: Define the function and the first principles formula. Let f(x) = x^2 + (3)/(2). The derivative from first principles is given by the formula: f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Calculate f(x+h). Substitute (x+h) into the function f(x): f(x+h) = (x+h)^2 + (3)/(2) Expand the term (x+h)^2: f(x+h) = x^2 + 2xh + h^2 + (3)/(2) Step 3: Calculate f(x+h) - f(x). Subtract f(x) from f(x+h): f(x+h) - f(x) = (x^2 + 2xh + h^2 + (3)/(2)) - (x^2 + (3)/(2)) f(x+h) - f(x) = x^2 + 2xh + h^2 + (3)/(2) - x^2 - (3)/(2) f(x+h) - f(x) = 2xh + h^2 Step 4: Form the difference quotient (f(x+h) - f(x))/(h). Substitute the expression from Step 3 into the difference quotient: (f(x+h) - f(x))/(h) = (2xh + h^2)/(h) Factor out h from the numerator: (h(2x + h))/(h) Cancel out h (since h ≠ 0 as h approaches 0): (f(x+h) - f(x))/(h) = 2x + h Step 5: Take the limit as h 0. Apply the limit to the simplified difference quotient: f'(x) = _h 0 (2x + h) As h approaches 0, the term h becomes 0: f'(x) = 2x + 0 f'(x) = 2x The derivative of x^2 + (3)/(2) with respect to x from first principles is 2x. Last free one today — make it count tomorrow, or type /upgrade for unlimited.