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myphysicslab – math refresher

here is a quick math refresher on calculus and trig, to help you enjoy the math behind the physics simulations at myphysicslab.

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)
ddt  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 and n are real non-zero constants. and h(t), g(t) are functions of t.

first consider powers of t. the general rule is ddt  t n = n t n − 1      for any n ≠ 0
here are some examples of derivatives of powers, using the above rule: ddt  t = 1
ddt  t 2 = 2 t
ddt  (t 3 + t 2 + t + 1) = 3t 2 + 2t + 1
ddt  1t = ddt  t −1 = − t −2 = −1t 2
here are some basic rules about derivatives: ddt  k = 0      (k = constant)
ddt  (k h(t)) = k  ddt h(t)      (k = constant)
ddt  (h(t) + g(t)) = ddt h(t) + ddt g(t)
ddt  (h(t) × g(t)) = h×g' + h'×g      the product rule
here are derivatives of some very important special functions ddt sin(t) = cos(t)
ddt cos(t) = −sin(t)
ddt e t = e t
ddt ln(t) = 1t      natural logarithm
the all-important chain rule lets us take the derivative of functions of functions (also called function composition): ddt h(g(t)) = h'(g(t)) × g'(t)      the chain rule
it's important to get good at using the chain rule. here are some examples of the chain rule in action: ddt  sin(h(t)) = cos(h(t)) h'(t)
ddt  sin(t 2) = 2 t cos(t 2)
ddt  e h(t) = h'(t) e h(t)
ddt  e k t = k e k t      (k = constant)
ddt  e t 2 = 2 t e t 2
ddt  ln(h(t)) = h'(t) ⁄ h(t)
ddt  ( 1 )  =   h'(t)
 h(t)  h(t)2
the quotient rule gives us the derivative of a ratio of functions:
ddt  ( h(t) )  =   g h'h g'      the quotient rule
g(t) g 2
using the chain rule and the product rule we can derive the quotient rule: ddt  h(t)g(t) = ddt  (h × 1g) = h' × (1g) + h × (g'g2) = (g h'h 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(x) = arctan(x) while tan2(x) = (tan(x))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. sin(−x) = −sin x
cos(−x) = cos x
tan(−x) = −tan x
sin x = cos(π2x)
cos x = sin(π2x)
sin(0) = 0
cos(0) = 1
sin(π2) = 1
cos(π2) = 0
sin(π) = 0
cos(π) = −1
sin(2) = −1
cos(2) = 0
sin(x + 2nπ) = sin x      n an integer
cos(x + 2nπ) = cos x      n an integer
the famous pythagorean theorem gives us the following identity cos2x + sin2x = 1
the sum of angles formulas are cos(x + y) = cos x cos y − sin x sin y
cos(xy) = cos x cos y + sin x sin y
sin(x + y) = sin x cos y + cos x sin y
sin(xy) = sin x cos y − cos x sin y
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