Question

Find a formula for the derivatives of the function g(x)=2x^2−3 using difference quotients.

How can I start this off?

Answer 1

start wid the limit definition...

Answer 2

lim f(a+h)-f(a)
h->0 ____________
h

??

Answer 3

so you need to find what f(a+h) is

Answer 4

well in this case, you're using g(x), so it would be g(x+h)

I'm using x in place of 'a' to keep the variable x in the final answer

Answer 5

g(x)=2x^2−3

g(x+h)=2(x+h)^2−3 ... replace ALL copies of x with x+h

now expand/simplify

Answer 6

ahh okay i see :)

so now 2(x+h)(x+h)-3 ?

so 2(x^2 + 2hx + h^2) -3 ?

Answer 7

keep going

Answer 8

2x^2 + 4hx + 2h^2 - 3 ?

Answer 9

good,

g(x+h) = 2x^2 + 4hx + 2h^2 - 3

Answer 10

is there more that i need to do? :o

Answer 11

now plug the expressions for g(x) and g(x+h) into

\[\Large \lim_{h\to 0} \frac{g(x+h) - g(x)}{h}\]

Answer 12

ahh okay..

so lim (2x^2 + 4hx + 2h^2 - 3) - (2x^2 + 4hx + 2h^2 - 3)
h->0 ______________________________________________________
not sure what h is :/

and is the top right? Or should it be - 2x^2 - 3 ?

Answer 13

no

Answer 14

you have g(x+h) - g(x) in the numerator

NOT g(x+h) - g(x+h)

Answer 15

lim (2x^2 + 4hx + 2h^2 - 3) - 2x^2- 3
h->0 ______________________________________________________
h

?

Answer 16

forgot that you just leave h as the variable hahaa

Answer 17

don't forget the parenthesis grouping

Answer 18

think of

g(x+h) - g(x)

as

[ g(x+h) ] - [ g(x) ]

Answer 19

lim (2x^2 + 4hx + 2h^2 - 3) - (2x^2- 3)
h->0 ____________________________________
h

is that better?

Answer 20

much better

Answer 21

now simplify that expression

Answer 22

lim 4hx
h->0 _________
h

lim 4x
h->0

?

Answer 23

or it just becomes lim h->0 = 4x ?

Answer 24

hmm not quite

Answer 25

darn:(

Answer 26

when you distribute the negative through, you get

(2x^2 + 4hx + 2h^2 - 3) - (2x^2- 3)

2x^2 + 4hx + 2h^2 - 3 - 2x^2 + 3

Answer 27

2x^2 + 4hx + 2h^2 - 3 - 2x^2 + 3 simplifies to ???

Answer 28

4hx + 2h^2
_____________
h

so 4x+2h ?

forgot the 2h^2!!

Answer 29

good, you factor out the h from the numerator, and then the pair of h's cancel leaving you with 4x + 2h

Answer 30

after that h in the denominator is gone, you can finally plug in h = 0 and evaluate

Answer 31

because you no longer have to worry about a division by zero error

Answer 32

ahh woo! :)

okay, so 4x+2(0)=4x ? so is taht the final answer? 4x ?

Answer 33

yep, 4x

Answer 34

\[\Large \lim_{h\to 0} \frac{g(x+h) - g(x)}{h} = 4x\]

when

\[\Large g(x) = 2x^2 - 3\]

Answer 35

ahh okay..

so now

g(4x)=2(4x)^2 - 3 ?

Answer 36

in other words, the derivative of g(x) = 2x^2 - 3 is g ' (x) = 4x

Answer 37

why are you plugging in 4x?

Answer 38

oh so you just leave it as 4x?

Answer 39

and that's the derivative for the equation?

Answer 40

once you get the derivative, you're done

Answer 41

ahh okay awesome!! thank you!!!:)

Answer 42

np