MyPhysicsLab - Math Refresher

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Here is a quick math refresher for those of you who once learned this stuff but haven't seen it for a while.

Calculus: Derivatives

The notation for the first derivative of a function x(t), with respect to the variable t, can be written as

x'

x'(t)

x(t)

These are all equivalent.  The notation for the second derivative is x'' or x''(t).
 
Here are some of the basic rules for calculating derivatives.  In the following k is a real constant and n is an integer constant. And f(t), g(t) are general functions of t.

k = 0
k t = k
k t² = 2 k t
k tⁿ = n k t^n-1
(t³+t²+t+1) = 3t²+2t+1
(1/t) = t^-1 = -t^-2 = -1/t²
k f(t) = k f(t)
(f(t)+g(t)) = f(t) + g(t) = f' + g'
f(t)g(t) = f*g' + f'*g   The product rule
f(g(t)) = f'(g(t)) g'(t)    The famous "chain rule"
sin(k t) = k cos(k t)
cos(k t) = -k sin(k t)
e^{k t} = k e^{k t}
ln(t) = 1/t   Natural logarithm
f(t)/g(t) = (g f' - f g')/g²   The quotient rule

Don't let that last one scare you, its easy to rederive if you use the product rule:

f(t)/g(t) = f*(1/g) = f'*(1/g) - f*(1/g²)*g' = (g f' - f g')/g²

Trig Identities

First, a note on some confusing notation: an exponent of -1 on a trig function means the inverse of that function (not the reciprocal!). Therefore
    tan^-1(a) = arctan(a)
while
    tan²(a) = (tan(a))²
The best way to get comfortable with trigonometry is to think in terms of the unit circle. Most of these identities then become obvious.  In the following a,b are real constants and n is an integer constant.

sin(-a) = -sin a
cos(-a) = cos a
tan(-a) = -tan a
cos²a + sin² = 1
cos(a+b) = cos a cos b - sin a sin b     sum of angles
sin(a+b) = sin a cos b + cos a sin b
sin a = cos(π/2 - a)
cos a = sin(π/2 - a)
sin(0) = 0
cos(0) = 1
sin(π/2) = 1
cos(π/2) = 0
sin(π) = 0
cos(π) = -1
sin(3π/2) = -1
cos(3π/2) = 0
sin(a + 2n π) = sin a
cos(a + 2n π) = cos a