Question

Difference Quotients Help. f(x)=1/x, f(x)-f(a)/x-a. I get how to plug in 1/x in the equation. Its the simplifying part that I stumble on.

Answer 1

\[ \frac{ f(x) - f(a)}{x-a} = \frac{1/x - 1/a}{x-a} = \frac{(a-x)/(xa)}{x -a }\]
Can you take it from there?

Answer 2

what did you do between step 2 and 3?

Answer 3

This:
\[ \frac{1}{x} - \frac{1}{a} = \frac{a - x}{ax} \]
That is, created a fraction with a common denominator

Answer 4

can you show that step? did you take 1/x times a/a or what?

Answer 5

You want a common denominator. That denominator has to be xa, because the denominators of 1/x and 1/a are x and a respectively.

So how do you write 1/x as
something/xa?

It is
1/x = a/ax

Likewise
1/a = x/ax

Thus

1/x - 1/a = a/ax - x/ax

Yes?

Answer 6

ok that makes sense.

Answer 7

Whats the easiest way to go from there?

Answer 8

So we have shown that
\[ \frac{f(x) - f(a)}{x-a} = \frac{(a-x)/xa}{x-a} = \frac{1}{xa} . \frac{a-x}{x-a} \]

Looks like a pretty obviously cancellation to be done there!

Answer 9

*obvious

Answer 10

ok thank you

Answer 11

The answer is -1/ax

Answer 12

Yes