Derivative

In finance, a derivative can refer to a derivative security.

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In linguistics and etymology, a derivative is a word which is "derived" from another.

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In mathematics, the derivative of a function is one of the two central concepts of calculus. The derivative of a function, at some point, is a measure of the rate at which that function is changing as an argument undergoes change. A derivative is the computation of the instantaneous slopes, of f(x) at every point x.

This corresponds to the slope of the tangent, to the graph of said function, at said point; the slopes of such tangents can be approximated by a secant. If the velocity of a car is given, as a function of time; then, the derivative of said function describes the acceleration of said car, as a function of time.

Differentiation can be used to determine the change which something undergoes, as a result of something else changing; if a mathematical relationship betweeen the two objects has been determined. A function is differentiable, at x, if its derivative exists at x; a function is differentiable, in an interval, if a derivative exists for every x within the interval. The derivative of f(x) is f '(x); also known as d/dx[f(x)], df/dx, D_x[ f ], the "derivative of f with respect to x", and "f prime of x",.

Newton's Difference Quotient

Derivatives are defined by taking the limit of a secant slope, as its two points of intersection (with f(x)) converge; the secant approaches a tangent. This is expressed by Newton's difference quotient; where h is Δx (the distance between the secant's points of intersection):

$f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

Since immediately substituting 0 for h results in division by zero, the numerator must be simplified such that h can be factored out and then canceled against the denominator. The resulting function, f '(x), is the derivative of f(x).

Example 1 Consider the graph of f(x) = 2x - 3. Should the reader have an understanding of algebra and the Cartesian coordinate system; then, the reader should be able to independently determine that this line has a slope of 2 at every point. Using the above quotient (along with an understanding of the limit, secant, and tangent) one can determine the slope at (4,5):

m (as h approaches 0) = [f(x + h) - f(x)] / h = [f(4 + h) - f(4)] / h = {[2(4 + h) - 3] - [2(4) - 3]} / h = (8 + 2h - 3 - 8 + 3) / h = 2h / h = 2

Example 2 Via differentiation, one can find the slope of a curve. Consider f(x) = x²:

m (as h approaches 0) = [f(x + h) - f(c)] / h = [(x + h)² - x²] / h = [x² + 2xh + h² - x²] / h = [2xh + h²] / h = 2x + h = 2x

For any point x; the slope (of f(x) = x² = f '(x) = 2x.

Example 3 Consider f(x) = √x:

m (as h approaches 0) = [f(x + h) - f(c)] / h = [√(x + h) - √x] / h = {[√(x + h) - √x][√(x + h) + √x]} / h[√(x + h) + √x] = [(x + h) - x] / h[√(x + h) + √x] = 1 / [√(x + h) + √x] = 1 / 2√x

Algebraic Manipulations

"Messy" limit calculations can be avoided in concrete cases because of differentiation rules which allow one to find derivatives via algebraic manipulations. One should not therefore infer that the definition of derivatives in terms of limits is unnecessary. Rather, that definition is the means of proving those "powerful differentiation rules".

Points on the graph of a function where the derivate is equal to zero are referred to as a stationary points.

Suppose we wish to find the derivative of a suitably smooth function, f say, at the point x. If we increase x by a small amount, which we'll call Δx, we can calculate f(x + Δx). An approximation to the slope of the tangent to the curve is given by (f(x + Δx) - f(x)) / Δx, which is to say it is the change in f divided by the change in x. The smaller the value Δx, the better approximation.

Mathematically, we define the derivative, denoted f '(x), to be the mathematical limit of this ratio as Δx tends to zero. Functions which possess a derivative are said to be differentiable, and finding the derivative is also called differentiation. Instead of using the tedious limit definition in order to find the derivative at a single point, it is also possible to find the derivative function f ' (also written as df/dx) which records all derivatives of f at all points.

Time Derivative

Arguably the most important application of calculus to physics is the concept of the time derivative -- the rate of change over time -- which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

• Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the time derivative of position.
• Acceleration is the time derivative of velocity.
• Jerk is the time derivative of acceleration.

Although "time derivative" can be written simply as "d/dt", it also has a special notation: a dot placed over the symbol for the object whose time derivative is being taken.

Rules for Evaluating Derivatives

The success of calculus stems from the surprising fact that f '(x) can be easily computed from an expression for f(x) using a small number of algebraic rules:

• Linearity: (a f + b g)' = a f ' + b g ' for all functions f and g and all real numbers a and b.
• Power rule: If f(x) = x^r with some real number r, then f '(x) = r x^r-1.
• Product rule: (f g)' = f ' g + f g' for all functions f and g.
• Quotient rule: (f / g)' = (f ' g - f g') / g²
• Chain rule: If f(x) = g(h(x)), then f '(x) = g'(h(x)) h'(x)
• Inverse functions and differentiation: If y = f(x) then x = f^-1(y). Assume that f(x) and its inverse are continuous and differentiable. For cases in which Δx ≠ 0 when Δy ≠ 0, dy/dx=1/(dx/dy)
• Derivative of one variable with respect to another when both are functions of a third variable: Let x = f(t) and y = g(t). Now Δy/Δx = (Δy/Δt)/(Δx/Δt) (Assume that Δx ≠ 0 when Δy ≠ 0). Hence, on letting Δt → 0, dy/dx = (dy/dt)/(dx/dt).
• Implicit differentiation: If f(x, y) = 0 be an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).

In addition, the derivatives of some common functions are useful to know. See the table of derivatives.

As an example, the derivative of f(x) = 2 x⁴ + sin(x²) - ln(x) e^x + 7 is f '(x) = 8 x³ + 2x cos(x²) - 1/x e^x - ln(x) e^x.

Multiple Derivatives

When the derivative of a function of x has been found, the result, being also a function of x, may be also differentiated, which gives the derivative of the derivative, or, as called, the second derivative. Similarly, the derivative of the second derivative is called the third derivative, and so on. If we were strictly to adhere to our notation, we should denote the several derivatives of y by:

${\frac {df}{dx}}$ ${\frac {d\left({\frac {df}{dx}}\right)}{dx}}$ ${\frac {d\left({\frac {d\left({\frac {df}{dx}}\right)}{dx}}\right)}{dx}}$

and so on.

In order to avoid so cumbersome a notation, the following symbols are often preferred:

${\frac {df}{dx}}$ ${\frac {d^{2}f}{dx^{2}}}$ ${\frac {d^{3}f}{dx^{3}}}$

and so on.

Using Derivatives to Graph Functions

Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all "critical points" (points at which the derivative of the function has determinant zero) are mapped to local extrema; some are so-called "saddle points". The Second Derivative Test is one way to evaluate critical points: if the second derivative of the function at the critical point is positive, then the point is a local minimum; if it is negative, the point is a local maximum; if it is neither, the point is either saddle point or part of a locally flat area (possibly still a local extremum, but not absolutely so). (In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension, at local extrema.)

Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side. Also, the supremum of a continuous function on an open and bounded domain will also be one of the local maxima; the infemum will be one of the local minima--this gives one an easy way to find the bounds of the function's range.

More Info

Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Mathematicians tend to speak the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.

The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.

In order to differentiate all continuous functions and much more, one defines the concept of distribution.

For differentiation of complex functions of a complex variable see also Holomorphic function.

See also: differintegral.