# Thread: Differentiation from first principles

1. ## Differentiation from first principles

I'm stick on differentiating, from first principles, the following function:

f(x) = (3x-1)/(x+2)

I plug that into:

[f(a+h) - f(a)]/h

and I get:

7h/(x+h+2)(x+2)

But when I use the quotient rule to do it, I end up with:

7/(x+2)^2

I've been trying to get rid of that bothersome h, but I just haven't found a way to do it! I have a feeling that I'm overlooking something really simple. Any help appreciated.

2. Well, you have to divide by h before you take the limit as h goes to zero, right?

3. It's probably easiest to simplify the fraction before trying to differentiate it.

$f(x) = \frac{3x - 1}{x + 2}$

$= \frac{3x + 6 - 7}{x + 2}$

$= \frac{3(x + 2)}{x + 2} - \frac{7}{x + 2}$

$= 3 - \frac{7}{x + 2}$.

Therefore $f(x + h) = 3 - \frac{7}{x + h + 2}$.

$f(x + h) - f(x) = \left(3 - \frac{7}{x + h + 2}\right) - \left(3 - \frac{7}{x + 2}\right)$

$= \frac{7}{x + 2} - \frac{7}{x + h + 2}$

$= \frac{7(x + h + 2) - 7(x + 2)}{(x + 2)(x + h + 2)}$

$= \frac{7x + 7h + 14 - 7x - 14}{(x + 2)(x + h + 2)}$

$= \frac{7h}{(x + 2)(x + h + 2)}$.

So $\frac{f(x + h) - f(x)}{h} = \frac{\frac{7h}{(x + 2)(x + h + 2)}}{h}$

$= \frac{7h}{h(x + 2)(x + h + 2)}$

$= \frac{7}{(x + 2)(x + h + 2)}$.

So $\lim_{h \to 0}\frac{f(x + h) - f(x)}{h} = \lim_{h \to 0}\frac{7}{(x + 2)(x + h + 2)}$

$= \frac{7}{(x + 2)(x + 2)}$

$= \frac{7}{(x + 2)^2}$.

4. Originally Posted by blackdragon190
I'm stick on differentiating, from first principles, the following function:

f(x) = (3x-1)/(x+2)

I plug that into:

[f(a+h) - f(a)]/h

and I get:

7h/(x+h+2)(x+2)
No. That "h" in the numerator should not be there.

But when I use the quotient rule to do it, I end up with:

7/(x+2)^2

I've been trying to get rid of that bothersome h, but I just haven't found a way to do it! I have a feeling that I'm overlooking something really simple. Any help appreciated.

5. Silly me! I got so caught up in playing around with the fraction that I forgot that I still had to divide by h. I just knew that it was something silly.

Thanks for the help

6. You're welcome for whatever I contributed. Have a good one!

7. Seems like I forgot about the limit too...doh..