Question

Find and simplify the difference quotient
[ f(x+h) - f(x) ] / h, h does not equal 0
f(x) = x^2

Answer 1

2x

Answer 2

answer is not 2x and i need an explanation

Answer 3

Ohh, sorry I thought it was when h tends to 0 xD

Answer 4

oh haha idk how to do it im confused on how to get the 2x

Answer 5

f(x) = x^2
f(x+h) = (x+h)^2
[ f(x+h) - f(x) ] / h = [ (x+h)^2 - x^2 ] / h = (x+h+x)(x+h-x) / h = (2x+h)(h) / h =
(2x+h)

Answer 6

As h->0, 2x+h -> 2x

Answer 7

wait how does (x+h)^2 - x^2 go to (x+h+x)(x+h-x)

Answer 8

There is an algebraic identity for the difference of two squares:

\(a^2 - b^2 = (a+b)(a-b) \)

Therefore, \((x+h)^2 - x^2 = (x+h+x)(x+h-h) = (2x+h)(x)\)

Answer 9

OHHHH okay

Answer 10

I meant \((x+h)^2 - x^2 = (x+h+x)(x+h-x) = (2x+h)(h)\)

Answer 11

so how do you get the ending x? i understand (x+h)(x+h) then idk the -x^2

Answer 12

typo in my reply before the last one which I corrected in the previous reply.

Answer 13

im reading the most recent one but i still dont understand the ending x?

Answer 14

There is a 'h' at the end not 'x'

Answer 15

in the second part

Answer 16

a^2 - b^2 = (a+b)(a-b)
(x+h)^2 - x^2 = ?
a = x+h, b = x
a + b = x + h + x = 2x + h
a - b = x + h - x = h
(a+b)(a-b) = (2x+h)(h)

Answer 17

okay i understand now, thank you so much

Answer 18

You are welcome.

You can also do this the long way:
(x+h)^2 = x^2 + 2xh + h^2
(x+h)^2 - x^2 = x^2 + 2xh + h^2 - x^2 = 2xh + h^2 = (2x+h)(h)