tckthomas wrote:there seems to be too much trig identities here! i can only remember the reciprocal ones, sine(x) = 1/cosecant(x), cosine(x) = 1/secant(x), tangent(x) = 1/cotangent(x)

You'll find most people won't remember them, just try to recognise when there might be one that could apply to an equation, then look it up. Although if you use them all the time, then you'll easily remember them.

tckthomas wrote:factorials? i think they are just too weird. is it impossible to derive a factorial if it isn't over another factorial?

If by derive, you mean differentiate, then that is impossible. A factorial is not a continuous function, which is a prerequisite to be differentiated; but be warned, not every continuous function is differentiable. A good example of a non-differentiable continuous function is |x|, where |-3| = 3, |0| = 0, |1| = 1. If you graph this function, it looks like to lines that meet sharply at the point (0,0); it is defined everywhere and doesn't "jump"*, therefore it is continuous. You see from the graph that coming from the positive x, dy/dx = 1; then coming from the negative x, dy/dx = -1. There is no real clear definition for dy/dx at x=0, and going from first principles doesn't work.

tckthomas wrote:implicit differentiation i think i get it. derive both sides with respect to x, then dy/dx magically appears, then put that on one side and done.

It is actually that when differentiating, in terms of x, a term of a function that involves y , you need to account for the fact that y changes differently to x. The rate at which y changes in terms of x sounds familiar... it is just dy/dx. So if you have the earlier function, the full calculation is:

x

^{2}+y

^{2}=r

^{2}d/dx(x

^{2}+y

^{2}) = d/dx(r

^{2})

d/dx(x

^{2})+d/dx(y

^{2}) = d/dx(r

^{2})

Now r is a constant, but y is a variable; so we have to differentiate y!

2*x + 2*y*(dy/dx) = 0

2*y*(dy/dx) = -2*x

dy/dx = -x/y

tckthomas wrote:chain rule/product rule/quotient rule, i need to find out when to use the chain rule. I just don't know why to use the chain rule when sin(3x^2+1)... can you tell me more?

The reason the chain rule exists is because it is impossible to differentiate the composition of two functions ie. one function inside another, ex. f(x) = sin(x), g(x) = 2*x

^{2}+1, f(g(x)) = sin(g(x)) = sin(2*x

^{2}+1).

In order to differentiate f(g(x)), we just differentiate f(x), leaving g(x) the same inside; then we work out g'(x) and multiply it onto the end. We can even have multiple chain rules:

f(x) = cos

^{2}(x

^{3}) = ( cos(x

^{3}) )

^{2}This is really three functions: g(x) = x

^{2}, h(x) = cos(x), i(x) = x

^{3}, f(x) = g(h(i(x)))

f'(x) = g'(h(i(x)))*h'(i(x))*i'(x)

f'(x) = 2*cos(x

^{3})*(-sin(x

^{3}))*(3*x

^{2})

* The mathematical definition for continuous functions is a bit complex, if you want to try look at wikipedia. Basically, as the distance between two x values decreases, then the distance between the corresponding y-values is also decreasing; also this difference will approach zero.

You really need to understand the formal definition of a limit if you want to understand this.

http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit#Precise_statement