Question

find dy/dx of y=1/x using (f(x+h) -f(x)) / h. I need step by step explaination. I keep getting really lost in all the fractions

Answer 1

again?

Answer 2

you really want to understand this and do it step by step?
we can if you like

Answer 3

i havent got an answer with explaination yet. i know the answer i just cant do the formula.

Answer 4

ok we can work it out slowly. lets start with
\[\frac{1}{3}-\frac{1}{2}\]

Answer 5

because the only part that requires any work is subtracting the fractions. i am not trying to insult your intelligence, it is just that if we do it for numbers we will see why it works here

Answer 6

or would you prefer if i just write the steps necessary?

Answer 7

Its ((1/x+h)-(1/x))/h but then what do you multiply top and bottom by?

Answer 8

nothing. we will actually subtract in the numerator

Answer 9

\[\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}\]

Answer 10

so
\[\frac{1}{x+h}-\frac{1}{x}=\frac{x-(x+h)}{(x+h)x}\]

Answer 11

therefore the numerator of the "difference quotient" is
\[\frac{-h}{(x+h)x}\]

Answer 12

now we want to divide this by h right? and when we divide by h we get
\[\frac{-1}{(x+h)x}\]

Answer 13

now you take the limit as h goes to zero by simply replacing h by 0 in the above expression and get
\[\frac{-1}{(x+0)x}=-\frac{1}{x^2}\]

Answer 14

any step not clear?

Answer 15

I've never heard of the b-a/ab. is that just something your supposed to know? is there no other way to do it?

Answer 16

that is how i subtract fractions. more genally it is
\[\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\] but in this case the numerators are 1. it is not some special formula, it is just how one subtracts fractions

Answer 17

that is why i wanted to start with
\[\frac{1}{3}-\frac{1}{2}\] so we remember it is
\[\frac{2-3}{3\times 2}=\frac{-1}{6}\]

Answer 18

likewise
\[\frac{1}{5}-\frac{1}{4}=\frac{-1}{20}\]

Answer 19

what happened to the orignial h in the denomenator?

Answer 20

once we got
\[\frac{-h}{(x+h)x}\] that was just the numerator of the "difference quotient"
we still had to divide by h

Answer 21

it was
\[\frac{f(x+h)-f(x)}{h}=\frac{\frac{-h}{(x+h)x}}{h}\]

Answer 22

when you divide by h, the h's cancel. that is why you end up with just
\[\frac{-1}{(x+h)x}\]

Answer 23

more clear now?