Two questions
Find the derivative of f(x)=
I got the derivative
Just confirming that's correct? I have to use the first principle rule to find it.
Second question is find the derivative of f(x) =
I believe the derivative should be
But when I work through the question (which I dont understand how to work through very well, because the example I was shown involved a 1 in the numerator not a 4)
I end up at
I'm not strong enough in latex to list these steps properly I dont think...
All of that, over h since the first principle is
I dont know how to simplify any farther correctly (or get a 3 tier fraction on LaTex), I've tried following along with my example I was shown but it doesn't work correctly I dont think.
Any input is greatly appreciated, hopefully the LaTex helps to make my equations and stuff more clear.
It's hard to answer your questions because you do not tell us what general principles of differentiation you know.
Anyway, starting from first principles
$f(x) = \dfrac{4}{x + 1} \implies f(x + h) = \dfrac{4}{x + h + 1} \implies f(x + h) - f(x) = \dfrac{4}{x + h + 1} - \dfrac{4}{x + 1} =$
$\dfrac{4(x + 1)}{(x + 1)(x + h + 1)} - \dfrac{4(x + h + 1)}{(x + 1)(x + h + 1)} =\dfrac{4x + 4 - 4x - 4h - 4}{(x + 1)(x + h + 1)} = \dfrac{- 4h}{(x + 1)(x + h + 1)}.$
With me to here?
$\therefore \dfrac{f(x + h) - f(x)}{h} = \dfrac{1}{h} * \dfrac{- 4h}{(x + 1)(x + h + 1)} = \dfrac{- 4}{(x + 1)(x + h + 1)}.$
$Now\ \displaystyle \lim_{h \rightarrow 0}(- 4) = - 4\ and\ \lim_{h \rightarrow 0}(x + 1)(x + h + 1) = (x + 1)(x + 1) = (x + 1)^2.$
$So\ \displaystyle \lim_{h \rightarrow 0}\dfrac{- 4}{(x + 1)(x + h + 1)} = \dfrac{\displaystyle \lim_{h \rightarrow 0}(- 4)}{\displaystyle \lim_{h \rightarrow 0}(x + 1)(x + h + 1)} = \dfrac{- 4}{(x + 1)^2}.$
There are of course simpler ways to do this.
thank you very much Jeff, basically you started with the inverse so ?
After sitting down for about an hour I got to the correct answer, still trying to figure out your steps but it will click after a few glances.
As for question one, I didn't do the coding correctly. I meant for question ones derivative to be which is the derivative of
I don't understand what you mean by inverse in this context.
There are two ways to find a derivative: (1) apply certain laws of derivatives, which is what you normally do after the first week or so of class, or (2) apply the basic definition of a derivative. I did the latter because I don't know what laws you have learned.
Anyway, the definitional approach actually gives you a procedure for finding a derivative.
Definition $f'(x) = \displaystyle \lim_{h \rightarrow 0}\dfrac{f(x + h) - f(x)}{h}.$
Procedure (from definition rather than laws)
Write down f(x).
Write down f(x + h)
Write down f(x + h) - f(x)
Simplify that difference algebraically.
Divide the simplified difference by h (or multiply by 1/h).
Find the limit of that quotient as h approaches 0 using the laws of limits.
That is exactly the procedure I used.
Thanks Jeff, I actually do this stuff through correspondence, or else I would just be asking my teacher (not to be rude, because I really do appreciate the help) I am working on learning a lot of laws in this lesson. I see what you did now so thats good.
As for the second question, yes is the equation