MHB Simplifying the Difference Quotient for f(x) = 1/x

AI Thread Summary
To simplify the difference quotient for f(x) = 1/x, start with the expression (f(x) - f(a)) / (x - a), which becomes (1/x - 1/a) / (x - a). This can be rewritten as ([a - x] / (ax)) / (x - a), leading to (a - x) / (ax(x - a)). Utilizing the identity a - x = -(x - a) allows for further simplification, resulting in the final answer of -1 / (ax). The process highlights the importance of algebraic manipulation in calculus.
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I have a problem:

f(x) = 1/x,

[f(x) - f(a)] / x- a

I am wondering how to approach this problem.

I have so far.

(1/x - 1/a) / (x-a)

([a-x] / xa) / (x-a)

How would I simplify this?

By the way, the answer is

-1 / ax
 
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Here's a re-write of your own approach using $\LaTeX$

$$\frac{f(x) - f(a)}{x - a} = \frac{\frac1{x} - \frac1{a}}{x - a} = \frac{\frac{a-x}{ax}}{x - a} = \frac{a - x}{ax(x-a)}$$

Can you use the fact $a - x = -(x-a)$?
 
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