Derivative

See noaseboar's writeup below for the definition. I might add that the nth derivative of a mapping f: X → Y can be regarded as living in the space of symmetric n-multilinear maps X → Y, which is naturally isomorphic to Hom(Symⁿ X, Y) where Symⁿ X is the nth symmetric power of X. See Geometric measure theory by H. Federer for a careful treatment of the multilinear algebra of geometric calculus, which can get tricky.

As the derivative is the instantaneous rate of change of one variable with respect to another, the derivative of position with respect to time is velocity, the derivative of velocity (wrt time) is acceleration, the derivative of acceleration (wrt time) is jerk.

f^I(x) = lim_x->a(f(x) - f(a))/(x - a)

There are also many formulas for finding derivatives of common functions without having to go through that whole limit nonsense.

When f(x) = xⁿ
f^I(x) = nx^n-1

When f(x) = c (Where c is a constant)
f^I(x) = 0

When f(x) = e^x
f^I(x) = e^x

When f(x) = a^x
f^I(x) = ln(a) * a^x

When f(x) = ln(x)
f^I(x) = 1/x

When f(x) = sin(x)
f^I(x) = cos(x)

When f(x) = cos(x)
f^I(x) = -sin(x)

When f(x) = tan(x)
f^I(x) = sec²(x)

When f(x) = tan^-1(x)
f^I(x) = 1/(1 + x²)

Also you can use the rules on functions like: f(x) = xⁿ + x or n^x/xⁿ thinking of it as several functions combined using the following rules:

(f(x) + g(x))^I = f^I(x) + g^I(x)

(f(x) - g(x))^I = f^I(x) - g^I(x)

(f(x) * g(x))^I = f(x) * g^I(x) + f^I(x) * g(x)

(f(x) / g(x))^I = ((g(x) * f^I(x)) - (f(x) * g^I(x)))/g(x)²

Some examples:
f(x) = sin(x) + x⁵
f^I(x) = cos(x) + 5x⁴

f(x) = x⁴2x⁵
f^I(x) = (x⁴10x⁴) + (4x³2x⁵)

            f(x+h) - A(h) - f(x)
lim         ---------------------------
||h|| -> 0            ||h||

Let L(X,Y) the normed space of continuous linear functions from X to Y.
If the derivative of f exists in an open neighborhood of x, then the derivative of the map Df: X -> L(X,Y), x |-> Df(x) might exist. It's called the second derivative of f in x, written as D²f(x).
If the second derivative exists in an open neighborhood of x, then the derivative of D²f : X -> L(X,L(X,Y)) might exist and is called the third derivative.
In fact the n-th derivative is the derivative of the function D^n-1f : X -> L(X,...L(X,Y))...)), x-> D^n-1f(x).
However these "stacked" spaces of linear functions L(X,...L(X,Y))..)) are difficult to use. Therefore one uses the fact that L(X,..L(X,Y)...) with n L's stacked is isometric to B(X,Y,n) is the space of n-linear continuous functions (Note: B(X,Y,n) is not canonical for this space, I just made it up)
The isomorphism is defined per: h of L(X,...L(X,Y)..) goes to g of B(X,Y,n) via g(x₁,...,x_n):= h(x₁)...(x_n).
So one takes as the n-th derivative the function Dⁿf(x₁)...(x_n) instead of Dⁿf(x₁) of L(X,...L(X,Y)..) (n-1 L's stacked)

The function f is called n times continuous differentiable in x iff the map Dⁿf: X -> B(X,Y,n), x |-> Dⁿf(x) is continuous (Note: this is not a linear map !)

Now comes the question: "What has this to do with the usual derivative of R¹ -> R¹ ?"
The derivative of R¹ is scalar. Multiplication with a scalar is the form of linear maps from R¹ to R¹. Set Df(x)(y) := f'(x)· y and you get the above form.

This definition allows you to differentiate in really sick spaces like function space, spaces of matrices etc.
The derivatives are quite difficult to determine there but some simple laws still hold:

• The derivative of a continuous linear map is the map itself at any point of X.
• The chain rule always holds: D(f(g))(x)(y) = (Df)(g(x))(Dg(x)(y)) where (Df)(g(x)) is the derivative of f at point g(x), the formula means: to get the image of y under D(f(g)) first apply Dg(x) on y and then apply (Df)(g(x)) ( f(g) is the function you get when you apply f to the images of g)

The derivative of a function of one variable, f(x), is another function f'(x). Geometrically, the derivative represents the slope of a line tangent to the graph of f at x.

The derivative is one of the fundamental concepts of the calculus, developed around the same time by both Sir Isaac Newton and Gottfried Wilhelm von Leibniz. Leibniz used a different notation to represent the derivative: "df/dx" which is read as "the derivative of f with respect to x."

The derivative of the derivative of a function is referred to as the second derivative (and this can go on to third, fourth, fifth, etc.).

Functions of more than one variable do not have a derivative, but have partial derivatives with respect to each independent variable. A partial derivative is obtained by treating all other independent variables as constants and then performing normal differentiation.

The slope of a secent line through two points, (x1,y1) and (x2,y2), on a the graph of f(x) is given by:

f(x2)-f(x1)
-----------
x2-x1

The difference in x between the two points is change in x or more commonly: deltaX. Thus, this formula can be rewritten as:

f(x1+deltaX)-f(x1)
------------------
deltaX


The tangent line is the same as the secent line, except there is no change in x. So if we take the limit of this expression as deltaX approaches 0, then we get a new function of x which represents the slope of a line tangent to f at x. This is the definition of the derivative.

A derivative is a contract between a pair of counterparties, due to be executed on some future date, apart from options, where the execution (exercise) is optional.

The Underlying

• equities. These are stocks and shares in individual companies
• bonds
• commodities, including gold, petroleum, sugar, coffee, bananas etc.

• exchange rate, such as USD -> EUR
• index, such as the Dow Jones industrial average, FTSE 100
• interest rate

Types of derivative

• futures

  Futures are an agreement to trade an underlying asset at a future date, based on today's price. Futures are exchange traded derivatives, and require payments of margin. Expiry of futures contracts happens on a 3 monthly cycle, usually March, June, September and December.

• Forwards

  Forwards are an OTC equivalent of futures. As such, the issuing bank can be flexible, making a tailor made contract. However, there is the need to provide collateral, and/or credit references. One popular type of forward is the forward rate agreement, or FRA, offering forwards on interest rates.

• Options

  Options are concerned with buying and selling the right to buy or sell an underlying. The right to buy is called a call option, and the right to sell is called a put option. Each option has two counterparties, the buyer of the option, the long party, and the seller (writer), the short party. There are four combinations:

  • buying the right to buy. Long on call.
  • selling the right to buy. Short on call.
  • buying the right to sell. Long on put.
  • selling the right to sell. Short on put.

  In exchange for the options contract, the long party pays a premium to the short party, much in the manner of insurance. The short party is the one which is exposed to the risk, as Nick Leeson demonstrated spectacularly.

  Options are traded both OTC and on stock exchanges.

• Swaps

  Similar to a playground scenario, a swap involves the exchange between counterparties of one underlying for another underlying. An everyday example of a swap transaction is exchanging a variable rate property mortgage for a fixed rate mortgage.

  Swaps are usually OTC derivatives. Sometimes they involve currencies, for instance USD may have a different exposure to risks than EUR. Also popular are interest rate swaps.

What derivatives are used for

However, derivatives can also be used for speculation, as Enron have demonstrated. Many countries require financial insitutions taking risks to account for their transactions on a daily basis, and to put hedging in place for risky transactions - as a legal requirement.

Source: Mastering Derivatives Markets - Francesca Taylor. Prentice Hall

                          f(x1 + δx) - f(x1)
                        ---------------------
                                  δx

                          f(x1 + δx) - f(x1)
       f|(x1) = limδx->0 ---------------------
                                  δx

                          (x1 + δx)n - x1n
       f|(x1) = limδx->0 ---------------------
                                  δx

where o(δx³) means terms where δx is present in at least the third power.

                          x1n + nx1n-1δx + (n(n-1)/2)x1n-2δx2 + o(δx3) - x1n
       f|(x1) = limδx->0 ----------------------------------------------------
                                                  δx
                         nx1n-1δx + (n(n-1)/2)x1n-2δx2 + o(δx3)
              = limδx->0 --------------------------------------
                                        δx
                        nx1n-1δx       (n(n-1)/2)x1n-2δx2        o(δx3)
              = limδx->0 -------   +   -----------------   +   -------
                           δx                 δx                 δx

f^|(x₁) = nx₁^n-1

De*riv"a*tive (?), a. [L. derivativus: cf. F. d'erivatif.]

Obtained by derivation; derived; not radical, original, or fundamental; originating, deduced, or formed from something else; secondary; as, a derivative conveyance; a derivative word.

Derivative circulation, a modification of the circulation found in some parts of the body, in which the arteries empty directly into the veins without the interposition of capillaries.

Flint.

-- De*riv"a*tive*ly, adv. -- De*riv"a*tive*ness, n.

© Webster 1913.

De*riv"a*tive, n.

1.

That which is derived; anything obtained or deduced from another.

2. Gram.

A word formed from another word, by a prefix or suffix, an internal modification, or some other change; a word which takes its origin from a root.

3. Mus.

A chord, not fundamental, but obtained from another by inversion; or, vice versa, a ground tone or root implied in its harmonics in an actual chord.

4. Med.

An agent which is adapted to produce a derivation (in the medical sense).

5. Math.

A derived function; a function obtained from a given function by a certain algebraic process.

⇒ Except in the mode of derivation the derivative is the same as the differential coefficient. See Differential coefficient, under Differential.

6. Chem.

A substance so related to another substance by modification or partial substitution as to be regarded as derived from it; thus, the amido compounds are derivatives of ammonia, and the hydrocarbons are derivatives of methane, benzene, etc.

© Webster 1913.