# integral of 1/x

This is a discussion on integral of 1/x within the A Brief History of Cprogramming.com forums, part of the Community Boards category; trig identities go as such, remember that f' = [f(x+h) - f(x)]/h as h ->0 sin(x+h) = sin x cos ...

1. trig identities go as such,

remember that f' = [f(x+h) - f(x)]/h as h ->0
sin(x+h) = sin x cos h + cos x sin h

therefore:

f' = lim (sin x cos h + cos x sin h - sin x )/h as h ->0

using the properties of limits:

f' = lim (sin x cos h - sin x)/h + lim cos x sin h/h as h->0

factor sin x:

lim sin x(cos h - 1)/h + lim cos x sin h/h as h->0

based on the fact that sin x and cos x have nothing to do with h(i.e. they are not involved in the limit):

sin x lim (cos h-1)/h + cos x lim sin h/h as h->0

computing the limit of (cos h - 1)/h and sin h/h as h -> 0 we have

sin x * 0 + cos x * 1 = cos x

EDIT: I know you can find out for yourself, just wanted to save you some trouble :P

and about chain rule, you are right, it is pretty intuitive:

if y = f(x)
and z = g(x)
and h(x) = f(g(x))

dy/dx = dy/dz * dz/dt

It is just that doing it this way tends to make people think of leibnix notation as a fraction, which in some ways it is, but most of the time it is better to think of it as just a notation.

2. Sandwich rule?

3. Sandwich rule
Sandwiching is the following, say if I have 3 functions:

f(g(h(x)))

the derivative of this is : f'(g(h(x))) * g'(h(x)) * h'(x)

4. Right... I can't say I had ever heard that term before.

5. lmao...I've never heard of it either...made me hungry though

6. Sandwiching is the following, say if I have 3 functions:

f(g(h(x)))

the derivative of this is : f'(g(h(x))) * g'(h(x)) * h'(x)
ive never heard that called the Sandwich rule...i usually here it called the Chain Rule.

7. I know, but I was just making a guess at it, as I honestly dont know what it is either....

8. sandwich rule == sandwich theorem

Maybe you've heard of it differently but it's where:

sin(h) / h == 1
lim h -> 0

9. Originally posted by Silvercord
sandwich rule == sandwich theorem

Maybe you've heard of it differently but it's where:

sin(h) / h == 1
lim h -> 0
Is that what you call sandwich rule??

I think i took it in Calculus but i ve never heard the word sandwich when i studied it

to be frank man the law you ve stated is derived from L'Hopital rule because if you substitute both sin(h) and h with zeros you will get 0/0
so in Hopital rule u differentiate both numerator and denominator by h so it will be cos(h)/1 where cos(0)=1
thus sin(h)/h =1

so it's L'hopital rule not sandwich one

10. no i re-read it in my book im right

11. Originally posted by Silvercord
no i re-read it in my book im right
No you aren't. What book are you reading? The only "Sandwich Rule" I've heard of is used to describe limits of a function by "sandwiching" it between two functions.

EDIT: The Squeezing (Sandwich) Theorem
The Actual Sandwich Theorem

12. i don't know too much about calc, but i found a site on this theorem, if it helps any...:

http://www.sosmath.com/calculus/limc.../limcon03.html

13. No you aren't. What book are you reading? The only "Sandwich Rule" I've heard of is used to describe limits of a function by "sandwiching" it between two functions.
well maybe it's not the only example of the sandwich/pinching theorem but i'm on page 127 of "Calculus, Graphical, Numerical, Algebraic" by "Finney, Thomas, Demana, Waits" and it uses sin(h) / h, and that's the only way I had ever seen it done until you posted your links.

14. thanks jverkoey, that stuff is actually in my calculus book. wonder why I never realized it.

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