Find and simplify the difference quotient (f(x+h) - f(x))/h for f(x) = 7/(3x - 1).

Step-by-step solution

1. 1

   Write f(x + h). Substituting into the denominator only:

   $f(x+h) = \frac{7}{3(x+h)-1} = \frac{7}{3x+3h-1}$

2. 2

   Set up the compound fraction.

   $\frac{f(x+h)-f(x)}{h} = \frac{1}{h}\left(\frac{7}{3x+3h-1} - \frac{7}{3x-1}\right)$

   The key is to combine the inner difference before dividing by $h$; otherwise nothing cancels.

3. 3

   Combine over a common denominator.

   $\frac{7(3x-1) - 7(3x+3h-1)}{(3x+3h-1)(3x-1)}$

   The numerator expands to $21x - 7 - 21x - 21h + 7 = -21h$ — every term except the $-21h$ cancels, which is the whole point of the manoeuvre.

4. 4

   Divide by h and cancel.

   $\frac{-21h}{h\,(3x+3h-1)(3x-1)} = \frac{-21}{(3x+3h-1)(3x-1)}$

5. 5

   Verify and take the limit. At $x = 1.3$, $h = 0.07$: the direct quotient is $-2.3284178$, and the formula gives $\tfrac{-21}{(3.11)(2.9)} = -2.3284178$. As $h \to 0$ the expression tends to $\tfrac{-21}{(3x-1)^2}$, which is indeed $\tfrac{d}{dx}\tfrac{7}{3x-1}$.