Here is a quick math refresher on calculus and trig, to help you enjoy the math behind the physics simulations at MyPhysicsLab.
The notation for the first derivative of a function x(t) , with respect to the variable t , can be written as
x'
x'(t)
^{d}⁄_{dt}
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
^{d}⁄_{dt} t ^{n} = n t ^{n − 1} for any n ≠ 0
Here are some examples of derivatives of powers, using the above rule:
^{d}⁄_{dt} t = 1
^{d}⁄_{dt} t ^{2} = 2 t
^{d}⁄_{dt} (t ^{3} + t ^{2} + t + 1) = 3t ^{2} + 2t + 1
^{d}⁄_{dt} ^{1}⁄_{t} = ^{d}⁄_{dt} t ^{−1} = − t ^{−2} = −^{1}⁄_{t 2}
Here are some basic rules about derivatives:
^{d}⁄_{dt} k = 0 (k = constant)
^{d}⁄_{dt} (k h(t)) = k ^{d}⁄_{dt} h(t) (k = constant)
^{d}⁄_{dt} (h(t) + g(t)) = ^{d}⁄_{dt} h(t) +
^{d}⁄_{dt} g(t)
^{d}⁄_{dt} (h(t) × g(t)) = h×g' + h'×g The product rule
Here are derivatives of some very important special functions
^{d}⁄_{dt} sin(t) = cos(t)
^{d}⁄_{dt} cos(t) = −sin(t)
^{d}⁄_{dt} e ^{t} = e ^{t}
^{d}⁄_{dt} ln(t) = ^{1}⁄_{t} Natural logarithm
The all-important chain rule lets us take the derivative of functions of functions (also called function composition):
^{d}⁄_{dt} h(g(t)) = h'(g(t)) × g'(t)
The chain rule
It is important to get good at using the chain rule. Here are some examples of the chain rule in action:
^{d}⁄_{dt} sin(h(t)) = cos(h(t)) h'(t)
^{d}⁄_{dt} sin(t^{2}) = 2 t cos(t^{2})
^{d}⁄_{dt} e ^{h(t)} = h'(t) e ^{h(t)}
^{d}⁄_{dt} e ^{k t} = k e ^{k t} (k = constant)
^{d}⁄_{dt} e ^{t2} = 2 t e ^{t2}
^{d}⁄_{dt} ln(h(t)) = h'(t) ⁄ h(t)
^{d}⁄_{dt} ( | 1 | ) = | − h'(t) |
h(t) | h(t)^{2} |
The quotient rule gives us the derivative of a ratio of functions:
^{d}⁄_{dt} ( | 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:
^{d}⁄_{dt} ^{h(t)}⁄_{g(t)} = ^{d}⁄_{dt} (h × ^{1}⁄_{g}) = h' × (^{1}⁄_{g}) + h × (^{−g'}⁄_{g2}) = (g h' − h g') ⁄ g^{2}
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
tan^{2}(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(^{π}⁄_{2} − x)
cos x = sin(^{π}⁄_{2} − x)
sin(0) = 0
cos(0) = 1
sin(^{π}⁄_{2}) = 1
cos(^{π}⁄_{2}) = 0
sin(π) = 0
cos(π) = −1
sin(^{3 π}⁄_{2}) = −1
cos(^{3 π}⁄_{2}) = 0
sin(x + 2 nπ) = sin x n an integer
cos(x + 2 nπ) = cos x n an integer
The famous pythagorean theorem gives us the following identity
cos^{2}x + sin^{2}x = 1
The sum of angles formulas are
cos(x + y) = cos x cos y − sin x sin y
cos(x − y) = cos x cos y + sin x sin y
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y − cos x sin y
This web page was first published April 2001.