PhysLab.net – Math Refresher

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

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)
^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's 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 ^{t ²} = 2 t e ^{t ²}
^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') ⁄ 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(^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(^3π⁄₂) = −1
cos(^3π⁄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(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