## anonymous 3 years ago Problem : Find the derivatives of the function using the limit of definition y=x^4

1. anonymous

still the same type of question? try going http://www.math.hmc.edu/calculus/tutorials/limit_definition/ they show you examples. just apply your question to how they solve it.

2. anonymous

It cant be that you were taught 2 other similar questions but still cant solve it.

3. anonymous

im confused about the exponent 4

4. anonymous

do you know how to do (a+b)^4?

5. anonymous

no

6. anonymous

its the binomial theorem or pascal triangle

7. anonymous

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 etc....

8. anonymous

for example just look at the pascal triangle for the coefficients (a+b)^2= 1a^2 + 2ab +1b^2= a^2 +2ab+b^2 can you guess whats (a+b)^3? and for (a+b)^4?

9. anonymous

note it can be like (x+y)^3 ? or (x+y)^4 ?

10. anonymous

so power of 3 we get the coefficients of 1 3 3 1 therefore the answer for (x+y)^3= x^3 +3x^2y +3 xy^2 + y^3 can you guess for (x+y)^4 ?

11. anonymous

the reason i want you to learn this is when you go solved for the derivative of y=x^4 you will get to find (x+y)^4

12. anonymous

13. anonymous

im here now to teach you the easier way of doing it

14. anonymous

ok for the power of 4 the coefficents inthe pascal triangle are 1 4 6 4 1 therefore (a+b)^4 = a^4 + 4a^3 b +6 a^2 b^2 +4 a b^3 + b^4 or (x+y)^4= x^4 +4x^3 y + 6 x^2 y^2 + 4x y^3 + y^4 did you see it ?

15. anonymous

yes

16. anonymous

now for practice y=x^2 $y+\Delta y=x +\Delta x$ $\Delta y=x+\Delta x - y$ $\Delta y= (x+\Delta x)^{2}-x ^{2}$

17. anonymous

can you try solving that first ?

18. anonymous

wait

19. anonymous

i try solving it

20. anonymous

21. anonymous

$\Delta y=f(x+\Delta x) - y$ $\Delta y=(x+\Delta x)^{2}- x ^{2}$ $\Delta y= x ^{2 }+ 2x \Delta x +\Delta x ^{2} -x ^{2}$ add or subtract we get $\Delta y= 2x \Delta x+\Delta x ^{2}$ now the derivative is $y'=f'(x)= \lim _{\Delta x ->0}\frac{ \Delta y }{ \Delta x }=$ $= \lim _{\Delta x->0}(\frac{ 2x \Delta x+\Delta x ^{2} }{ \Delta x })$ try to cancell something in there

22. anonymous

23. anonymous

ok good can you try it if its y=x^4 ? do the same process

24. anonymous

okay

25. anonymous

26. anonymous

hmm to make it easier lets make delta x=h so that delta y=(x+h)^4 - x^4 try solving that first

27. anonymous

I cant follow my professor's teaching. Good to know that u are here to help me .thank you so much. you are a great﻿ help! i need to do more assignments about this so i will go out now .

28. anonymous

29. anonymous

haha im wrong

30. anonymous

do the process i gave you there, and you will arrive at the correct answer ... :D

31. anonymous

i wish i have a tutor like u hahaha .

32. anonymous

remember this ?. (x+y)^4= x^4 +4x^3 y + 6 x^2 y^2 + 4x y^3 + y^4 make y an h here so that (x+h)^4= x^4 +4x^3 h + 6 x^2 h^2 + 4x h^3 + h^4

33. anonymous

anyway thx a lot

34. anonymous

goodbye!

35. anonymous

ok just read it through here again and learn from it good luck and have fun now :D

36. anonymous

ok

37. anonymous

ok just messege me here and if you have prob i will go and give you an advise or correct if you have something done incorrectly ... :D