Question

Use the definition f'(a)=lim h->0 (f(a+h)-f(a))/h
to find the derivative of f(x)=(1/x) at x=3. Please be detailed so I can clearly see the steps I can use for similar problems.

Answer 1

find f(x+h)= 1/(x+h), when x=3 , this works out to 1/(3+h)
f(x+h)-f(x)=1/(3+h)-1/3

that should give you a hint

Answer 2

Well, that is what I did. Then I got ((1/3+h)-(1/3))/h
The I simplified that to (1/3 + 1/h -1/3)/h , the 1/3 -1/3 cancel out, leaving (1/h)/h or (1/h) x (1/h) = (1/h)^2. What I am confused about is h -> 0, so now I plug 0 into (1/h)^2, and I know you cannot devide by 0, which makes me think I did something wrong or that the answer is simply 0. Am I correct? If not could you please show me the remaining steps, please?

Answer 3

You cant write 1/(3+h) = 1/3 + 1/h

Answer 4

thats the reason yu are getting errors

Answer 5

But even if I don't write 1/(3+h) as 1/3+1/h, I still have to plug in 0 into the equation, right? So ((1/3+h)-1/3)/h = (1/3-1/3)/0 = 0/0. Would that be 0 or would that be 1? Since anything divided by anything is 1? Or am I still wrong?

Answer 6

*What I meant by anything / anything is a number divided by itself.

Answer 7

1/(3+h) - 1/3=(3-3-h)/3(3+h)
so you have
(-h)/(3)(3+h)

now your limit changes to lim -h
-------------
h->0 3*(3+h)*h

Answer 8

check the steps i have suggested, pardon me for any errors

Answer 9

I'm sorry, I'm confused. Is there any easier way? :(

Answer 10

unfortunately NO! , the reason is you need to use the limit definition of derivative

Answer 11

Could you do the whole problem through so I can see how to properly do it? This problem is limiting me from finishing the rest of my homework and correctly understand what I'm supposed to do.

Answer 12

cancel out the -h and the h in denominator
you are left with -1/ (9+3h) , now evaluate the limits put h=0, you get -1/9 as the answer which IS THE ONLY CORRECT ANSWER

alternatively if you want to verfiy the derivative of 1/x the answer of that using the x^n formula is -1/x^2 which verifies the work

Answer 13

let me look this through

Answer 14

there is no mistake, the only thing i am worried is text appearing as garbled due to unicode format

Answer 15

text if fine, i just need my brain to understand it now. ill let you know if i have any more questions :)

Answer 16

*text is fine

Answer 17

Take your time