for the function f(x)= 3/(2x-1), determine the difference quotient and simplify
To find the difference quotient for a function, we need to consider the expression (f(x+h) - f(x)) / h. Let's calculate it step by step for the given function f(x) = 3/(2x-1):
Step 1: Replace f(x) with its expression:
f(x) = 3/(2x-1)
Step 2: Substitute f(x+h) in place of f(x) and simplify:
f(x+h) = 3/[2(x+h)-1]
Step 3: Calculate the numerator by subtracting f(x) from f(x+h):
Numerator = f(x+h) - f(x)
= [3/[2(x+h)-1]] - [3/(2x-1)]
Step 4: Simplify the expression obtained in step 3:
Numerator = [3(2x-1) - 3(2(x+h)-1)] / [(2(x+h)-1)(2x-1)]
= [6x - 3 - 6x - 6h + 3] / [(4x + 2h - 1)(2x-1)]
= [-6h] / [(4x + 2h - 1)(2x-1)]
Step 5: Divide the numerator by h to complete the difference quotient:
Difference Quotient = Numerator / h
= [-6h / h] / [(4x + 2h - 1)(2x-1) / h]
= -6 / [(4x + 2h - 1)(2x-1)]
So, the difference quotient for the function f(x) = 3/(2x-1) is -6 / [(4x + 2h - 1)(2x-1)].