Turn on thread page Beta
 You are Here: Home >< Maths

# Differentiate from first principles? watch

1. Hey guys, I have been set 5 questions for homework, and I know how to different and all, but I don't know how to differentiate "from first principles", so can anybody help to use this method to answer these questions for me step by step? Would be really appreciated

1) y = x^2 + 2
2) y = x^2 - x
3) y = 4x
4) y = x - 1
5) y = x^3

Many thanks
2. For all those just work out

[y(x+h) - y(x)]/h

simplify, and then set h to be zero
3. Differentiation from first principles means that for each of the functions, you calculate (where ) and then observe what happens as .

I'll get you started on the first of your problems, and then you can do the rest. You have , and so we need to look at

Expanding the (x+h)² bracket on the numerator gives

...and simplifying reveals that this is equal to

Clearly as h tends to zero, this tends to .

Now you do the rest!
4. (Original post by adil12)
Hey guys, I have been set 5 questions for homework, and I know how to different and all, but I don't know how to differentiate "from first principles", so can anybody help to use this method to answer these questions for me step by step? Would be really appreciated

1) y = x^2 + 2
2) y = x^2 - x
3) y = 4x
4) y = x - 1
5) y = x^3

Many thanks
If you're differentiating a function from first principle you ned to express it's derivative as follows:

and then manipulate it until you get the correct answer.
So for the first one, , therefore . Put them into that above expression and manipulate it until you don't have the fraction with a denominator which tends to zero.
5. (Original post by adil12)
Hey guys, I have been set 5 questions for homework, and I know how to different and all, but I don't know how to differentiate "from first principles", so can anybody help to use this method to answer these questions for me step by step? Would be really appreciated

1) y = x^2 + 2
2) y = x^2 - x
3) y = 4x
4) y = x - 1
5) y = x^3

Many thanks
You have to use the definition of differentiation:

Edit: 4 good replies in a minute. Speedy stuff guys :P
6. (Original post by nuodai)
Differentiation from first principles means that for each of the functions, you calculate (where ) and then observe what happens as .

I'll get you started on the first of your problems, and then you can do the rest. You have , and so we need to look at

Expanding the (x+h)² bracket on the numerator gives

...and simplifying reveals that this is equal to

Clearly as h tends to zero, this tends to .

Now you do the rest!
I see what you did here, clear and productive
Thanks a lot! Can you tell me how I can give you rep?
7. (Original post by PhyMath)
You have to use the definition of differentiation:

Edit: 4 good replies in a minute. Speedy stuff guys :P
Wow you guys replied so fast all with the same good answers, thanks to all for help And thanks mate, got it now
8. (Original post by Farhan.Hanif93)
If you're differentiating a function from first principle you ned to express it's derivative as follows:

and then manipulate it until you get the correct answer.
So for the first one, , therefore . Put them into that above expression and manipulate it until you don't have the fraction with a denominator which tends to zero.
Thanks mate, solid stuff
9. (Original post by RichE)
For all those just work out

[y(x+h) - y(x)]/h

simplify, and then set h to be zero
Thanks for the help man! Helped out fast and well
10. (Original post by RichE)
For all those just work out

[y(x+h) - y(x)]/h

simplify, and then set h to be zero
Set h to be zero?!

That's ******* appalling, that completely detracts from the whole idea behind calculus.

Newton/Liebniz would be turning in his grave.
11. (Original post by RichE)
For all those just work out

[y(x+h) - y(x)]/h

simplify, and then set h to be zero
Set h to be zero?!

That's ******* appalling, that completely detracts from the whole idea behind calculus.

Newton/Liebniz would be turning in his grave.
12. (Original post by adil12)
I see what you did here, clear and productive
Thanks a lot! Can you tell me how I can give you rep?
Pleasure Rep can be given by clicking the image at the top-right of a post which has the thumbs up/down in it.
13. (Original post by nuodai)
Pleasure Rep can be given by clicking the image at the top-right of a post which has the thumbs up/down in it.
Repped! By the way can you help me with the second question? I can do the rest but I don't know how to go about this one
14. (Original post by adil12)
Repped! By the way can you help me with the second question? I can do the rest but I don't know how to go about this one
If you can show some working, we'll be able to give you a better hint.
15. (Original post by turnand)
Set h to be zero?!

That's ******* appalling, that completely detracts from the whole idea behind calculus.

Newton/Liebniz would be turning in his grave.
I said it would work for those - which it does - and which is pretty much how Newton did it originally - which rather makes your post (C) Bishop Berkeley
16. (Original post by Farhan.Hanif93)
If you can show some working, we'll be able to give you a better hint.
Alright mate, I used the general formula and then got:
((x+h)^2 - x) - (x^2 - x)/h
= x^2 + 2hx + h^2 - x - x^2 + x /h
= 2x + h, and since h tends to be 0, answer is 2x, which I know is wrong because if I differentiate with powers I know it should be 2x-1 as the answer, where have I gone wrong?
17. (Original post by adil12)
Alright mate, I used the general formula and then got:
((x+h)^2 - x) - (x^2 - x)/h
= x^2 + 2hx + h^2 - x - x^2 + x /h
= 2x + h, and since h tends to be 0, answer is 2x, which I know is wrong because if I differentiate with powers I know it should be 2x-1 as the answer, where have I gone wrong?
If then
18. (Original post by adil12)
Alright mate, I used the general formula and then got:
((x+h)^2 - x) - (x^2 - x)/h
= x^2 + 2hx + h^2 - x - x^2 + x /h
= 2x + h, and since h tends to be 0, answer is 2x, which I know is wrong because if I differentiate with powers I know it should be 2x-1 as the answer, where have I gone wrong?
in this case...
19. (Original post by nuodai)
If then
But that's what I done no?
20. (Original post by adil12)
But that's what I done no?
The means "not equal to", but you subbed it in as if it were equal to it. It should have been as in Farhan's post (#18).

Turn on thread page Beta

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: November 3, 2012
Today on TSR

### University open days

1. University of Bradford
Wed, 25 Jul '18
2. University of Buckingham
Psychology Taster Tutorial Undergraduate
Wed, 25 Jul '18
3. Bournemouth University
Clearing Campus Visit Undergraduate
Wed, 1 Aug '18
Poll
Useful resources

## Make your revision easier

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

Can you help? Study help unanswered threads

## Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE