Question

Find the derivative of f directly from the limit definition of the derivative. f(x)=3/x

Answer 1

When they say to do it that way, you will have to set up the limit of a difference quotient or something along those lines.

Answer 2

\[\frac{ f(a-h)-f(a) }{ h }\]
Right?

Answer 3

As h goes to 0.

Answer 4

\[\lim_{h \rightarrow 0}\frac{ f(a-h)-f(a) }{ h }\]

Answer 5

Yep. That is one of the two ways to state the limit definition.

Answer 6

Plug them in and solve bro

Answer 7

I just don't really understand how to use the difference quotient. I didn't understand it in algebra either :(

Answer 8

It probably wasn't a good idea to close the question then

Answer 9

and all it is really is just plugging stuff in and solving

Answer 10

Remember function composition? \((f\circ g)(x)\) where you put one function into another? This is the same thing. You put \((a-h)\) into \(f(x)\) and the result goes where \(f(a-h)\) is.

Answer 11

we'll walk you though it if you want.. we got mccormick here :P

Answer 12

But yea just replace the x with a's and a-h's where needed. It's just like mccormick said

Answer 13

That would be great

Answer 14

There is only one x. maybe I am confused by simplicity? That is what I am hoping for.

Answer 15

It does not matter if it s 1 x or 50. It gets the same process.

Answer 16

It does not matter if it s 1 x or 50. It gets the same process.

If I say to evaluate \(x=6\), you do \(f(6)=\frac{3}{6}\implies f(6)=\frac{1}{2}\). Well, this time we want to evaluate \((a-h)\) so \(f(a-h)=\) what? Once you have that, you put it into te right place on the difference quotent. Then you do the same thing with just a. Then you have fun solving it!

Answer 17

hmm... some of that got repeated by accident. LOL

Answer 18

LOL I just did it too

Answer 19

I was trying to say maybe you could show me how you would find the derivative of 5/x using the limit definition? I am having a tough time computing that into what I need to do.

Answer 20

I may have just figured it out

Answer 21

You got this bro!

Answer 22

The limit definition of a derivative is stated one of two ways:
\[\lim_{h \rightarrow 0}\frac{ f(a-h)-f(a) }{ h }\]or
\[\lim_{x \rightarrow a}\frac{ f(x)-f(a) }{ x-a }\]Usually the first is easier.

Answer 23

Isn't the second one used for finding it at a certain point?

Answer 24

It can be used for when it becomes 0 as well. In fact, the first comes from the second.

Answer 25

So: \[\frac{ \frac{ 3 }{ a-h }-\frac{ 3 }{ a } }{ h }\]? Then slove from here?

Answer 26

ALMOST! You need to keep the limit part on there until you evaluate the limit.

Answer 27

Perfct, thank you!

Answer 28

But yea you're getting it, good job!

Answer 29

And if you did it right the answer should be -3/x^2

Answer 30

or -3/a^2

Answer 31

Yah, all the algebra to get it where:\[\lim_{h \rightarrow 0}\frac{ \frac{ 3 }{ a-h }-\frac{ 3 }{ a } }{ h }\]won't have 0 for a denominator, then it should simplify to something like what Luigi0210 is showing. When they say "limit definition" the goal is to have you show all the work between.

Answer 32

Don't worry eventually you'll learn a trick on how to find the derivative without having to do all this.. work

Answer 33

I have the correct answer, I just didn't get there the correct way. I avoid the difference quotient whenever possible! :)

Answer 34

See, quotients get no love! They are friendly creatures by their nature, but people see that bar in the middle and think it is dangerous! It starts with the fear of fractions and grows throughout the mathematical career.

Answer 35

LOL, it's true!