Use the definition of a derivative to find $f'(x)$ when $f(x) = \frac{x}{(x+1)}$

Question

Use the definition of a derivative to find $f'(x)$ when $f(x) = \frac{x}{(x+1)}$ the definition being: $$\lim_{x \to c} \frac{f(x)−f(c)}{x - c}$$

I did, and I got up to Don's second step as shown in the answer. However I'm not sure how he was algebratically able to get to his third step, which is what stopped me. — Bygonaut, Nov 21 '13 at 19:27

DonAntonio · Answer 1 · 2013-11-21T19:37:43.150

6

$$\frac{f(x)-f(c)}{x-c}=\frac{\frac x{x+1}-\frac c{c+1}}{x-c}=\frac1{(x+1)(c+1)}\xrightarrow[x\to c]{}\frac1{(c+1)^2}$$

edited Nov 21 '13 at 19:37

answered Nov 21 '13 at 19:23

DonAntonio

(+1) Concise. Just $f(c)$ in the numerator would be more correct, if I'm correct... – Johnny Westerling Nov 21 '13 at 19:27
Thanks, @JohnnyWesterling: edited. And only $;f(c);$ in the numerator is correct...:) – DonAntonio Nov 21 '13 at 19:28
Of course. You're welcome :) – Johnny Westerling Nov 21 '13 at 19:29
Thank you Don, I am curious as to how you got to the third step algebratically? – Bygonaut Nov 21 '13 at 19:30
@Bygonaut, simplify first the numerator in the second step and you shall see it clearly... – DonAntonio Nov 21 '13 at 19:31
Well, I'm quite embarassed now. Thank you. – Bygonaut Nov 21 '13 at 19:34
Welcome to the "Oh, damn: how in the world didn't I see that before!" club, @Bygonaut. I'm honorary dictator of the club. – DonAntonio Nov 21 '13 at 19:35
Well @DonAntonio I think the derivative is $\frac{1}{(c+1)^2}$! – P.. Nov 21 '13 at 19:36
And you're right, @P.. : there was a typo there when I was writing down the answer and I misled myself and wrote it again at the end. Thanks, edited. – DonAntonio Nov 21 '13 at 19:38

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