Question

Use the lim x->a f(x)-f(a)/x-a
to find the derivative of f(x)= (x+3)/(3x+1) at the point x=1

Answer 1

The derivative formula is:
\[\frac{d}{dx}f(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-f(x)}{h}}\]
So lets substitute in \(f(x)=\frac{x+3}{3x+1}\)
\[\lim_{h\rightarrow 0}{\frac{f(x+h)-f(x)}{h}}=\lim_{h\rightarrow 0}{\frac{\frac{x+h+3}{3x+3h+1}-\frac{x+3}{3x+1}}{h}}=\lim_{h\rightarrow 0}{\frac{\frac{(x+h+3)(3x+1)-(x+3)(3x+3h+1)}{(3x+3h+1)(3x+1)}}{h}}\]
\[=\lim_{h\rightarrow 0}{\frac{3x^2+x+3xh+h+9x+3-(3x^2+9x+3xh+9h+x+3)}{h(3x+3h+1)(3x+1)}}\]
\[=\lim_{h\rightarrow 0}{\frac{3x^2+x+3xh+h+9x+3-3x^2-9x-3xh-9h-x-3}{h(3x+3h+1)(3x+1)}}\]
\[=\lim_{h\rightarrow 0}{\frac{h+3-9h-3}{h(3x+3h+1)(3x+1)}}=\lim_{h\rightarrow 0}{\frac{-8h}{h(3x+3h+1)(3x+1)}}\]
\[=\lim_{h\rightarrow 0}{\frac{-8}{(3x+3h+1)(3x+1)}}=\frac{-8}{(3x+1)(3x+1)}=\frac{-8}{(3x+1)^2}\]

Answer 2

that is one way, also can use
\[\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\]

Answer 3

Now we have:
\[\frac{d}{dx}f(x)=\frac{-8}{(3x+1)^2}\]
Lets evaluate it at x=1:
\[\left.\frac{-8}{(3x+1)^2}\right|_{x=1}=\frac{-8}{(3+1)^2}=\frac{-8}{4^2}=-\frac{1}{2}\]

Answer 4

less algebra if you plug in the numbers first i think, but either way will work

Answer 5

Ahh I see. Haha I self taught myself one way but those two methods both work beautifully as well. I just like to derive the derivative first, to get a general answer, then simplify. Personal preference haha

Answer 6

\[f(1)=1\]so a start is
\[\frac{\frac{x+3}{3x+1}-1}{x-1}\] then clear the complex fraction by multiplying top and bottom by \(3x+1\) to get
\[\frac{x+3-(3x+1)}{x-1}\] clean up the numerator as
\[\frac{-2x+2}{x-1}\] factor as
\[\frac{-2(x-1)}{x-1}\] then cancel and you are done

Answer 7

hmm maybe i made a mistake, because i got a different answer...

Answer 8

Oh that is wonderful!! So do you just substitute the x value into the f(x) equation to get the a value?

Also, thank you @KeithAfasCalcLover for a superbly detailed answer!!!

Answer 9

oooh i see my mistake damn

Answer 10

should be \(\frac{x+3-(3x+1)}{(x-1)(3x+1) }\)

Answer 11

@qwerter you can always compute the derivative first, plug in the numbers second
once you get to the "quotient rule" this will be really really easy, you won't have to do it by hand

Answer 12

But in terms of knowing what to plug in for the a value, how do I figure that out?

Answer 13

you are asked for the derivative at \(x=1\) so in this case you would find \(f'(1)\)

Answer 14

No problem @qwerter. "One way to skin a cat"!

Answer 15

lol

Answer 16

@satellite73 But if we find the derivitive at x=1, we get -.5