derivative

—derivatively, adv. —derivativeness, n.

/di riv"euh tiv/, adj.

1. derived.

2. not original; secondary.

n.

3. something derived.

4. Also called derived form. Gram. a form that has undergone derivation from another, as atomic from atom.

5. Chem. a substance or compound obtained from, or regarded as derived from, another substance or compound.

6. Also called differential quotient; esp. Brit., differential coefficient. Math. the limit of the ratio of the increment of a function to the increment of a variable in it, as the latter tends to 0; the instantaneous change of one quantity with respect to another, as velocity, which is the instantaneous change of distance with respect to time. Cf. first derivative, second derivative.

7. a financial contract whose value derives from the value of underlying stocks, bonds, currencies, commodities, etc.

[1400-50; late ME derivatif < LL derivativus, equiv. to L derivat(us) (see DERIVATION) + -ivus -IVE]

* * *

In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function.

The first derivative of a function is a function whose values can be interpreted as slopes of tangent lines to the graph of the original function at a given point. The derivative of a derivative (known as the second derivative) describes the rate of change of the rate of change, and can be thought of physically as acceleration. The process of finding a derivative is called differentiation.

* * *

▪ mathematics

      in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations (differential equation). In general, scientists observe changing systems (dynamical systems (analysis)) to obtain the rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions.

 Geometrically, the derivative of a function can be interpreted as the slope of the graph of the function or, more precisely, as the slope of the tangent line at a point. Its calculation, in fact, derives from the slope formula for a straight line, except that a limiting (limit) process must be used for curves. The slope is often expressed as the “rise” over the “run,” or, in Cartesian terms, the ratio of the change in y to the change in x. For the straight line shown in the figure—>, the formula for the slope is (y₁ − y₀)/(x₁ − x₀). Another way to express this formula is [f(x₀ + h) − f(x₀)]/h, if h is used for x₁ − x₀ and f(x) for y. This change in notation is useful for advancing from the idea of the slope of a line to the more general concept of the derivative of a function.

 For a curve, this ratio depends on where the points are chosen, reflecting the fact that curves do not have a constant slope. To find the slope at a desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, in general, the ratio will represent only an average slope between the points, rather than the actual slope at either point (see figure—>). To get around this difficulty, a limiting (limit) process is used whereby the second point is not fixed but specified by a variable, as h in the ratio for the straight line above. Finding the limit in this case is a process of finding a number that the ratio approaches as h approaches 0, so that the limiting ratio will represent the actual slope at the given point. Some manipulations must be done on the quotient [f(x₀ + h) − f(x₀)]/h so that it can be rewritten in a form in which the limit as h approaches 0 can be seen more directly. Consider, for example, the parabola given by x². In finding the derivative of x² when x is 2, the quotient is [(2 + h)² − 2²]/h. By expanding the numerator, the quotient becomes (4 + 4h + h² − 4)/h = (4h + h²)/h. Both numerator and denominator still approach 0, but if h is not actually zero but only very close to it, then h can be divided out, giving 4 + h, which is easily seen to approach 4 as h approaches 0.

      To sum up, the derivative of f(x) at x₀, written as f′(x₀), (df/dx)(x₀), or Df(x₀), is defined as

if this limit exists.

       differentiation—i.e., calculating the derivative—seldom requires the use of the basic definition but can instead be accomplished through a knowledge of the three basic derivatives, the use of four rules of operation, and a knowledge of how to manipulate functions.

* * *