If x=f(t), then dy/dx = (dy/dt)/(dx/dt)

dy/dx*dx/dy=1 or dx/dy = 1/(dy/dx)

d(logx)/dx = 1/x

d(Sinx)/dx = Cosx

d(Cosx)/dx = -Sinx

d(tanx)/dx = Sec²x

d(cotx)/dx = -Cosec²x

d(secx)/dx = secxtanx

d(cosecx)/dx = -CosecxCotx

In the last Ricky post there is a little mistake:
1. lim a f(x)=a m, and
4. lim f(x)/g(x)= m/n , n != 0.
There won't be any eroors if, instead of:
"Let lim f(x)=m and lim g(x)=n",
we write:
"Let lim f(x)=n and lim g(x) = m".

//After the correction remove this

Hold it, the limit of tan 1/x as x approaches zero is pi/2? That doesn't seem right.

Shouldn't it be, limit of arctan 1/x as x approaches zero equals pi/2?

limit as x approaches 0 of tan(1/x) is undefined.

Same goes with arctan.

Some Important Expansions:-

for -1<x<1

for every positive value of x<1.

This isn't rigourous, but more of a way to think about it.  limit 1/x as x goes to 0 from the positive side is positive infinity.  So what we really want to find is the limit of arctan(x) as x goes to infinity, which is pi/2.

But what about when we approach 1/x from the negative side?  Then it's negative infinity, and thus, we want to find the limit of arctan(x) as x approaches negative infinity.  That's -pi / 2. 

Since the limit from the left is not the limit from the right, the limit does not exist.

For a clear way to see it, just use a graphing calculator.  Or just do arctan(1/.000000001) and arctan(1/-000000001).

Last edited by Ricky (2006-04-11 05:25:59)

Maclaurin/Taylor Series

Exponentials and Logarithms

Trigonometric Functions

See bottom of post for definition of B_n.

See bottom of post for definition of B_n.

See bottom of post for definition of E_n.

See bottom of post for definition of B_n.

Hyperbolic Functions

See bottom of post for definition of B_n.

See bottom of post for definition of B_n.

See bottom of post for definition of E_n.

See bottom of post for definition of B_n.

B_n: The Bernoulli Numbers

The Bernoulli numbers B_n are a sequence of special rational numbers. Their derivation is outside the scope of this thread, but an explanation may be offered elsewhere upon sufficient demand. The first few Bernoulli numbers are(note that for odd n other than 1, B_n = 0):

E_n: The Euler Numbers

The Euler numbers E_n are a sequence of special numbers. Their derivation is outside the scope of this thread, but an explanation may be offered elsewhere upon sufficient demand. The first few Euler numbers are(note that for all odd n, E_n = 0):

Last edited by Zhylliolom (2006-08-07 20:51:43)