Mathematical Techniques and Formulae:

A1. Mathematical Techniques and Formulae

1. Functions
A simple function is an equation or formula with a single variable quantity one side of the "=" symbol. An example is:
y = ax + b
Let us say that in this case there are two variable quantities y and x, and the other quantities, a and b, do not vary. Formulae of this type can be represented by the general statement:
y = f(x)
which should be read: "y is a function of x", or "y depends on x".
y is known as the dependent variable, because its value is dictated by x.
x is known as the independent variable, because it can take on arbitrary values.

The full definition for a function is as follows:
"A function is a rule relating one set to another; such that if we take one element of one set (xn say) and apply the rule, we obtain a unique element of the other set (yn say).

In general, y may depend on any number of variables. We can express this situation by means of a comma-separated list:
y = f(p, q, r, s, t)
where p, q, r, s and t are independent variables.

2. Differentiation:
The derivative of a function is its rate of change with respect to one of its variables. It is a number which tels us how much the dependent variable will change for a given change in an independent variable. For the simple case y=f(x), the derivative is written dy/dx and should be understood to mean "the amount by which y changes when x is changed by an infinitesimal (i.e., infinitely small) amount". It is necessary to define derivatives in terms of infinitesimal changes because the relationship between y and x is not necessarily linear, and the numerical value of dy/dx will vary as x varies. Hence, in general, the process of differentiation gives a new function, which gives the numerical derivative for a particular value of x when that value of x is inserted.

For the simple linear function y=ax + b:
dy/dx = a
this tells us that a change in x will give 'a' times as much change in y. The constant 'b' makes no difference to the rate of change and so does not appear in the derivative. Only coefficients associated with (i.e., operated on by) the independant variable make a contribution.

Any quantity raised to the power of zero is 1. If we look at what happened when we differentiated y=ax, we see that differentiation subtracted 1 from its power, i.e.:
if y = ax¹, then dy/dx = ax⁰

The general rule for differentiation is:

if y = axⁿ then dy/dx = anx^n-1

Differentiation is a simple additive process. If y can be written as a number of functions of x added together (an arithmetic series), i.e.:
y = f1(x) + f2(x) + f3(x) + . . . . .
then
dy/dx = df1(x)/dx + df2(x)/dx + df3(x)/dx + . . . . .

It is however, not always possible to break a formula down into a series of terms of the form
axⁿ. In such cases we must apply special rules.

Function of a function:
If y is a function of z and z is a function of x, i.e.:

if y = f(z) and z = f(x)
then
dy/dx = (dy/dz) (dz/dx)

The product rule:

if y = u v
where both u and v are functions of x
then
dy/dx = v(du/dx) + u(dv/dx)

The Quotient rule:
By using the product rule and the function of a function rule we obtain:

If y = N/D
where both N and D are functions of x
then
dy/dx = [D(dN/dx) - N(dD/dx)] / D²

Special functions:

if y = Sin(ax) then dy/dx = a Cos(ax)

if y = Cos(ax) then dy/dx = -a Sin(ax)

Since Tan(ax) = Sin(ax)/Cos(ax), we may use the quotient rule to obtain:

if y = Tan(ax) then dy/dx = a / Cos²(ax)

The exponential is a special function which is equal to its own rate of change.

if y = e^x then dy/dx = e^x

if y = Loge(x) then dy/dx = 1/x

Partial differentiation:
If the value of a function is dependent on more than one independent variable, we may define a partial derivitave as the rate of change of the function with respect to one of its independent variables, all other variables being held constant. A partial derivative is denoted by the special symbol '∂' (known as 'partial d', 'curly d' or 'Jacobi's delta'), i.e.:
∂y/∂x is the rate of change of y with respect to x when all other variables are treated as constants.
The partial differential symbol (first used in 1770 by the Marquis de Condorcet) corresponds most closely to the cursive (joined-up) or italic form of the Cyillic letter 'de' or "dey" as it appears in some (but not all) fonts. Hence, one way to vocalise ∂y/∂x is "dey y by dey x"

Maxima and Minima:
If a function (e.g., a resonance curve) has a peak or a trough, the gradient of the function at the tip of the peak or trough is 1 (level) and its rate of change at this point is 0. Hence, we can find the peaks and troughs in a function by applying the condition:
∂y/∂x = 0
If a function has more than one peak or trough, there will be more than one solution for ∂y/∂x=0. In general, the degree (highest power of x) of the derivative will be the same as the number of peaks and troughs. E.g., for an electrical system with a single resonant frequency, there will be a positive and a negative frequency solution for f0. Hence the derivative with respect to frequency of the function which describes the resonance curve will be a quadratic equation.

Analytical solutions exist for all equations up to degree 4 (quartic). Beyond that, no general solutions exist and numerical methods (matrix diagonalisation) may be required.

3. Exponents and Logarithms:
>>> article to be inserted
>>> explain how to use differentiation rules above
>>> conversion between bases

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