Question

Find a formula for the derivatives of g(x) = 2x^2-3 using difference quotients?

Answer 1

So they want you to use the definition of the derivative :D which is kind of a pain in the butt... XD

remember that one or no? :D

Answer 2

yea f(x+h) -f(x) / h

Answer 3

yah, so you take the "limit" of the difference quotient, letting h approach 0. and that will give you an equation of the line tangent to g(x) at any point. (in other words, it will give you the derivative)..
setting up the problem is the tricky part :) need help with that?

Answer 4

yea thats where I am screwing up, with all the algebra and putting the equ together

Answer 5

\[g(x)=2x^2 -3 \rightarrow g(x+h)=2(x+h)^2 -3\]

Answer 6

\[\frac{ g(x+h)-g(x) }{ h } = \frac{ 2(x+h)^2-3 -[2x^2-3] }{ h }\]

Answer 7

setup look ok? c:

Answer 8

yep I got this far

Answer 9

expanding out the binomial will give us..\[\frac{ 2x^2+4xh+2h^2-3-2x^2+3 }{ h }\]

Answer 10

ok I can't expand the binomial correctly

Answer 11

ok ignore the 2 for a moment :) so its the square of the first term + 2 times the product of the terms in the middle + the square of the 2nd term
\[(x+h)^2=(x+h)(x+h)=x^2+xh+xh+h^2=x^2+2xh+h^2\]

Answer 12

I wasn't ignoring the 2. :)

Answer 13

wait ok I need to see how to put the 2 back into the equ

Answer 14

\[2(x+h)^2=2(x^2+2xh+h^2)\]

Answer 15

what is this called so I can get specific practice on this stuff, just factoring?

Answer 16

i think it's called "expanding binomials". its a binomial since it has two terms. and you multiply them out, expanding or something XD

Answer 17

ok cool

Answer 18

ok then can I factor a 2 from the numerator?

Answer 19

if you get time, you should check out pascal's triangle. there is a really important connection between binomials and his silly triangle :D it makes it much much easier to expand 3rd degree binomials and higher

Answer 20

i dont htink you want to factor out a 2 D: since you have some nice cancellations

Answer 21

oops I forgot about the rest of the equ
ok Pascals triangle , got it thanks!

Answer 22

ok I get 4xh+2h^2 /h

Answer 23

ah yes sounds good so far! :)

Answer 24

got disconnected from the web

Answer 25

ok so factor 2h from top and get 4x+h?
the book answer is 4x

Answer 26

ok good, you got to answer of 4x+h, ONLY NOW should you take the limit, allow h to approach 0.
we don't run into problems anymore since we're no longer dividing by h.