Question

If f(x)=x^3 , simplify the expression f(x+h)-f(x)/h

Answer 1

\[\Large \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}\]

Answer 2

Can you plug in the numbers?

Answer 3

He only need to simplify expression, not finding limit.

Answer 4

He doesn't need the derivative? So what exactly does he want to do?
Linear approximation at some x=a? or what?

looks like he needs the derivative.

Answer 5

Simplify the expression...?

Answer 6

#1 - You must learn your Order of Operations

f(x+h)-f(x)/h MEANS \(f(x+h) - \dfrac{f(x)}{h}\) and you will have a tough time simplifying that.

#2 - Substitute the expression and simplify it.

[f(x+h)-f(x)]/h MEANS \(\dfrac{f(x+h) + f(x)}{h}\) and you will have ta chance to succeed at the exercise.

You have \(f(x) = x^{3}\) - First substitute!

Answer 7

Yes just simplify.

Answer 8

seriously, @HELP_ME! ?

Answer 9

ok I help you.

What do you have so far?

Answer 10

@geerky42 I need your help

Answer 11

Well all of you have been slightly confusing so I dont have anything as of now.

Answer 12

Ok, refresh your brain, forgot what you read.

You are given that \(f(x) = x^3\), right? And you were asked to simplify \(\dfrac{f(x+h)-f(x)}{h}\).

Because \(f(x)=x^3\), you have \(f(x+h) = (x+h)^3\) which is \( (x+h) (x+h) (x+h)\)

Now substitute and simplify.

Answer 13

Since you decided to make the warning public, I choose to defend it publicly.

It was just a warning. Try to learn from it and be angry somewhere else. There IS no need to argue with the question or to swear at the teachers.

Can we get back to learning mathematics?

Answer 14

Okay I am sorry

Answer 15

So now we have (x+h)(x+h)(x+h)
------------
h
so we can cross cancel one of the h's and that leaves us
(x)(x+h)(x+h)
------------
1
which we can turn into
(x)(x+h)^2
???? did I do it?

Answer 16

Well, you should have \(\dfrac{ (x+h) (x+h) (x+h)-x^3}{h}\)

First, try to expand \( (x+h) (x+h) (x+h)\)

Answer 17

Follow these steps in which I used that that
\[
(a + b)^3 a^3 + 3 a^2 b + 3 a b^2 + b^3
\]


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

Answer 18

Typo
\[
(a + b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3
\]

Answer 19

This limit above will help you prove that the derivative of \(x^3\) is \( 3 x^2 \)

Answer 20

I am only trying to simplify though.

Answer 21

Read my post above the answer is
\[
h^2 + 3 h x + 3 x^2
\]

Answer 22

Are you positive? and could I get a second opinion from someone? I am just confused.

Answer 23

I am positive 100%

Answer 24

I cam confirm that answer is \(3x^2+3xh+h^2\)

Answer 25

What part are you confused with? we can explain more

Answer 26

You should try to redo all the steps in my answer on your own to learn this stuff

Answer 27

Okay I thank you both, but I was simply confused on combining the x and the h but I guess that you foiled?

Answer 28

basically, yeah.
\(=~(x+h)x^2+(x+h)2xh+(x+h)h^2\)

Simplify further and combine like terms.