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)
^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(t²) = 2 t cos(t²)
^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)		g²

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(^{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

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

This web page was first published April 2001.

previous next