Final answer:
The derivative of the function f(x) = 1/(2x) - 3 is calculated using the limit definition of the derivative to be -1/(4x^2).
Step-by-step explanation:
To find the derivative of the function f(x) = 1/(2x) - 3 using the definition of the derivative, we apply the limit process:
The derivative, f'(x), can be found using the following limit:
f'(x) = lim(h→0) [(f(x + h) - f(x)) / h]
Plugging the function into this formula:
f'(x) = lim(h→0) [ (1/((2(x + h))) - 3 - (1/(2x) - 3) ) / h ]
Simplify the expression inside the limit:
f'(x) = lim(h→0) [ (1/(2(x + h)) - 1/(2x)) / h ]
After finding a common denominator and simplifying further, we take the limit as h approaches 0:
f'(x) = -1/(4x^2)
This is the derivative of the function f(x).