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## lgbasallote Group Title A tip on how to become good in Differential Calculus: MEMORIZE as LITTLE as possible, and learn to MANIPULATE as MUCH as possible. one year ago one year ago

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1. mathslover Group Title

give MEDALS as much as possible :P just joking

2. lgbasallote Group Title

Here's a FEW of the things that are ESSENTIAL to differential calculus: $\frac{d}{dx} (x^n) = nx^{n-1}$ $\frac{d}{dx} (e^x) = e^x$ $\frac{d}{dx} (\ln x) = \frac 1x$ $\frac{d}{dx} (\sin x) = \cos x$ $\frac{d}{dx} (\cos x) = -\sin x$ Then of course, the product and quotient rules: product rule: $\frac{d}{dx}(uv) = udv + vdu$ quotient rule: $\frac{d}{dx} (\frac uv) = \frac{vdu - udv}{v^2}$ Then, five of the trigonometric formulas: $\sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \sin \beta \cos \alpha$ $\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \pm \sin \alpha \sin \beta$ $\sin^2 + \cos^2 = 1$ $\tan^2 + 1 = \sec^2$ $1 + \cot ^2 = \csc^2$ With these, and a little manipulation, you can survive Differential Calculus. However, of course, as you go on with the course, you use other formulas as well and you'll just remember it. For example $\frac{d}{dx} \sin h x = \cosh x$ However, if you can't remember it, you can always figure it out using the basic information I wrote above. This tip was taken from a true Calculus master (who happened to be a summa cum laude), and has been proven effective by hundreds of students whom he have taught.

3. mathslover Group Title

that will be beneficial for me .. thanks lgba

4. lgbasallote Group Title

welcome

5. Yahoo! Group Title

Chain Rule , ........ etc ??

6. hartnn Group Title

POINT NOTED.

7. lgbasallote Group Title

yes.. i forgot chain rule.. $\frac{d}{dx} (u) = u'du$

8. lgbasallote Group Title

it goes like that right?

9. lgbasallote Group Title

Note in everything i wrote, u and v stand for FUNCTIONS of X

10. lgbasallote Group Title

however, like i said...the things i wrote above are the most essential

11. Eyad Group Title

GooD work.

12. Vaidehi09 Group Title

oh. ill keep those points in mind. thanks!

13. goformit100 Group Title

Thanks from me too... they are really benifitiel ... i'll keep it in my mind... thanks again

14. across Group Title

To be totally honest with you, remembering that$\frac{df}{dx}=f'(x)=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$is much better than memorizing all of that up there, but I'm a masochist, so, yeah. :P

15. lgbasallote Group Title

yeah...that's probably the least possible lol

16. ParthKohli Group Title

When I hadn't started with Calculus, that $$\Delta x$$ actually intimidated me. Then I learnt that you could just call that $$h$$. haha

17. ParthKohli Group Title

$f'(x) = {\lim_{h \to 0} \;\;{f(x + h) - f(x) \over h}}$

18. ParthKohli Group Title

@lgbasallote Dude, you're a hero of OpenStudy. I look forward to seeing more tips from ya :) and believe me, I really needed them!

19. Kathi26 Group Title

$f \prime(x)=\lim_{x \rightarrow x _{0}}\frac{ f(x)-f(x_{0}) }{ x-x _{0} }$

20. ParthKohli Group Title

LGB, can you please write some tips for integration as well? It really gets messy when you start Calculus II and Integration by Parts!

21. Vaidehi09 Group Title

yea...for integration by parts...its all good as long as u remember those formulae! we need tips for when u dont!

22. Kathi26 Group Title

$\int\limits_{a}^{b}f(x)dx=F(a)-F(b)=-\int\limits_{b}^{a}f(x)dx$ $\frac{ d }{ dx }(\int\limits_{a}^{f(x)}g(x)dx)=\frac{ d }{ dx }(G(f(x)-G(a))=f \prime(x) \times g(f(x))$ $=\frac{ d }{ dx }G(f(x))$ $\int\limits_{}^{}\frac{ 1 }{ x }dx=\ln(x)$ $\int\limits_{}^{}\frac{ f \prime(x) }{ f(x) }dx=\ln(f(x))$ $\int\limits_{}^{}e^{x}dx=e^{x}$ $\int\limits_{}^{}f \prime(x)\times e ^{f(x)}dx=e^{f(x)}$ $\int\limits_{}^{}\sin(x)dx=-\cos(x)$ $\int\limits_{}^{}\cos(x)dx=\sin(x)$ $\int\limits_{}^{}f \prime(x) \sin(f(x))dx=-\cos(f(x))$ $\int\limits_{}^{}f \prime(x) \cos(f(x))dx=\sin(f(x))$

23. across Group Title

@ParthKohli, pop quiz! Evaluate$\int\frac{2x}{x^2+6}\,dx.$:)

24. ParthKohli Group Title

O Lord, please help me figuring out the technique I should use here. Trig substitution?

25. ParthKohli Group Title

No... not trig substitution

26. Vaidehi09 Group Title

jst substitute x^2 = u

27. ParthKohli Group Title

Yeah, right!$du = 2xdx$

28. Kathi26 Group Title

It's ln(x^2+6). You can use the f'(x)/f(x)case.

29. ParthKohli Group Title

Woohoo!$\int {1 \over u + 6}du$

30. hartnn Group Title

x^2+6=u will be more beneficial.

31. ParthKohli Group Title

$\int (u + 6)^{-1}du$

32. Vaidehi09 Group Title

ah yes...@hartnn

33. ParthKohli Group Title

Well, okay... yes, I did learn in calculus II about the root technique thingy

34. Vaidehi09 Group Title

then it'll just be integration of 1/u

35. ParthKohli Group Title

$\int {1 \over u}du$$\ln|u|$$\ln|x^2 +6|$?

36. ParthKohli Group Title

Oh well, nope.

37. ParthKohli Group Title

I turned $$dx$$ to $$du$$.

38. ParthKohli Group Title

oops

39. ParthKohli Group Title

$u = x^2 + 6$$du = 2xdx$

40. Vaidehi09 Group Title

nope. that was the ryt answer. that + C.

41. ParthKohli Group Title

Protect me from Calculus, O Heaven!

42. hartnn Group Title

your answer was correct ln |x^2+6| + c

43. ParthKohli Group Title

$\int {1 \over u}du$$\ln|x^2 + 6| + C$I always slip the + c off.

44. ParthKohli Group Title

I need more practice. Ugh!

45. hartnn Group Title

parth u learning limits also ??

46. ParthKohli Group Title

Already learnt, yes.

47. hartnn Group Title

ok, i was gonna give a tutorial on that....and definite integration...which u want first ?

48. ParthKohli Group Title

I'd like a tutorial on the applications of definite integrals!

49. hartnn Group Title

*noted

50. lgbasallote Group Title

....wow

51. ParthKohli Group Title

... wow * 2

52. Kainui Group Title

I think it should be noted that the quotient rule is kind of meaningless to remember when you have the product rule already. Division is the same thing as multiplying. For instance 5x/(3x^2+2x)=5x*(3x^2+2x)^-1

53. ParthKohli Group Title

That's @TuringTest's intuition too!

54. lgbasallote Group Title

but remember @Kainui i did not put in chain rule in what i wrote

55. Kainui Group Title

The less you memorize, the better. I pretty much consider the power, chain, and product rule all in one rule these days to condense it further, it just usually turns out to be 1 and not important. An example of this might be: $2x^3=2y^0(x)^3$ Well the derivative of the outside is 6x and multiplied by the derivative of the inside, which is 1. The derivative of the coefficient is also 0 so you end up adding 0 as well. $d/dx[2y^0(x)^3]=6y^0x^2(d/dx(x))+0=6x^2$

56. Kainui Group Title

Also a note about the integration by parts is a great thing to include here along with chain rule lol. They are both essential in my mind.

57. lgbasallote Group Title

this is differential calculus....how the heck did integration by parts come in o.O

58. Kainui Group Title

Yeah, good point. Another helpful thing is to think about what the graph looks like, it will help you visualize what the derivative will be. This is particularly useful for optimization problems where you are making a graph where the highest point is the thing you want, and taking the derivative and setting it equal to 0 to find that peak.