DERIVATIVE : Comprehensive Encyclopedia

The simplest type of derivative is the derivative of a real-valued function of a single real variable. It has several interpretations:

This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many generalizations of the derivative.

The remainder of this article discusses only the simplest case (real-valued functions of real numbers).

Differentiation and differentiability

Differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "dee why by dee ex" or "dee why over dee ex". The form "dee why dee ex" is also used conversationally, although it may be confused with the notation for element of area.

Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. One precise way to define the derivative is as a limit:

A function is differentiable at a point x if the above limit exists (as a finite real number) at that point; a function is differentiable on an interval if it is differentiable at every point within the interval.

As an alternative, the development of nonstandard analysis in the 20th century showed that Leibniz's original idea of the derivative as a ratio of infinitesimals can be made as rigorous as the formulation in terms of limits.

If a function is not continuous at x, then there is no tangent line and the function is not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there, as in the case of the function y=|x|, which has a kink at x=0. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.

The derivative of a function f at x is a quantity which varies if x varies. The derivative is therefore itself a function of x; there are several notations for this function, but f' is common.

The derivative of a derivative, if it exists, is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on. A function may have zero, a finite number, or even an infinite number of derivatives.

Newton's difference quotient

The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line.

Tangent line at (x, f(x))

Secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h)).

To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is

${f(x+h)-f(x)\over h}.$

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:

$f'(x)=\lim_{h\to 0}{f(x+h)-f(x)\over h}.$

Tangent line as limit of secants.

If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x.

One cannot obtain the limit by substituting 0 for h, since it will result in division by zero. Instead, one must first modify the numerator to cancel h in the denominator. This process can be long and tedious for complicated functions, and many short cuts are commonly used which simplify the process.

Examples

Consider the graph of $f (x) = 2 x - 3$ . Using algebra and the Cartesian coordinate system, one can independently determine that this line has a slope of 2 at every point. And that is indeed the result one finds using the derivative. The slope at (4,5) is, using the above quotient:

$f'(4)\,$	$= \lim_{h\rightarrow 0}\frac{f(4+h)-f(4)}{h}$
	$= \lim_{h\rightarrow 0}\frac{2(4+h)-3-(2\cdot 4-3)}{h}$
	$= \lim_{h\rightarrow 0}\frac{8+2h-3-8+3}{h}$
	$= \lim_{h\rightarrow 0}\frac{2h}{h} = 2.$

The derivative and slope are equivalent. Consider $f (x) = x 2$ :

$f'(x)\,$	$= \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$
	$= \lim_{h\rightarrow 0}\frac{(x+h)^2 - x^2}{h}$
	$= \lim_{h\rightarrow 0}\frac{x^2 + 2xh + h^2 - x^2}{h}$
	$= \lim_{h\rightarrow 0}\frac{2xh + h^2}{h}$
	$= \lim_{h\rightarrow 0}(2x + h) = 2x.$

For any point x, the slope of the function $f (x) = x 2$ is $f'(x) = 2 x$ .

Notations for differentiation

Lagrange's notation

The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

$f'(x) \;$	for the first derivative,
$f''(x) \;$	for the second derivative,
$f'''(x) \;$	for the third derivative, and in general
$f^{(n)}(x) \;$	for the nth derivative, ( $n \ge 0$ ).

Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Gottfried Leibniz. For the function whose value at x is the derivative of f at x, we write:

$\frac{d\left(f(x)\right)}{dx}.$

With Leibniz's notation, we can write the derivative of f at the point a in two different ways:

$\frac{d\left(f(x)\right)}{dx}\left.{\!\!\frac{}{}}\right|_{x=a} = \left(\frac{d\left(f(x)\right)}{dx}\right)(a).$

If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as:

$\frac{dy}{dx}.$

Higher derivatives are expressed as

$\frac{d^n\left(f(x)\right)}{dx^n}$ or $\frac{d^ny}{dx^n}$

for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:

$\frac{d \left(\frac{d \left( \frac{d \left(f(x)\right)} {dx}\right)} {dx}\right)} {dx}$

which we can loosely write as:

$\left(\frac{d}{dx}\right)^3 \left(f(x)\right) = \frac{d^3}{\left(dx\right)^3} \left(f(x)\right).$

Dropping brackets gives the notation above.

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$

(In the formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In non-standard analysis, however, they can be viewed as infinitesimal numbers that cancel.)

Newton's notation

Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name:

$\dot{x} = \frac{dx}{dt} = x'(t)$

$\ddot{x} = x''(t)$

and so on.

Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives, and then, only to denote derivatives with respect to time, as opposed to other types of variables.

Euler's notation

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator:

$D_x f(x) \;$	for the first derivative,
${D_x}^2 f(x) \;$	for the second derivative, and
${D_x}^n f(x) \;$	for the nth derivative, provided n > 2

This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable:

$D u \;$	for the first derivative,
$D^2 u \;$	for the second derivative, and
$D^n u \;$	for the nth derivative, provided n > 2

Euler's notation is useful for stating and solving linear differential equations.

Critical points

Points on the graph of a function where the derivative is undefined or equal to zero are called critical points or sometimes stationary points (where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minimum and maximum can only occur at critical points or endpoints. This is related to the extreme value theorem.

Physics

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 derivative (with respect to time) of an object's position.
Acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position.

For example, if an object's position on a line is given by

$x(t) = -16t^2 + 16t + 32 , \,\!$

then the object's velocity is

$\dot x(t) = x'(t) = -32t + 16, \,\!$

and the object's acceleration is

$\ddot x(t) = x''(t) = -32 . \,\!$

If the velocity of a car is given, as a function of time, then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.

Rules for finding the derivative

In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules.

Constant rule: The derivative of any constant is zero.
- Constant multiple rule: If c is some real number; then, the derivative of $c f (x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below).
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 $f (x) = x r$ , for some real number r; $f'(x) = r x r - 1$ .
Product rule: $(f g)' = f' g + f g'$ for all functions f and g.
Quotient rule: $(f / g)' = (f' g - f g') / (g 2)$ unless g is zero.
Chain rule: If $f (x) = h (g (x))$ , then $f'(x) = h'(g (x)) g'(x)$ .
Inverse function: If the function $f (x)$ has an inverse $g (x) = f - 1 (x)$ , then $g'(x) = 1 / f'(f - 1 (x))$ .
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 $d y / d x = (d y / d t) / (d x / d t)$ . This is the chain rule in the Leibniz notation.

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

As an example, the derivative of

$f(x) = x^4 + \sin (x^2) - \ln(x) e^x + 7\,$

$f'(x) = 4 x^{(4-1)}+ \frac{d\left(x^2\right)}{dx}\cos{x^2} - \frac{d\left(\ln {x}\right)}{dx} e^x - \ln{x} \frac{d\left(e^x\right)}{dx} + 0$

$= 4x^3 + 2x\cos {x^2} - \frac{1}{x} e^x - \ln(x) e^x.$

The first term was calculated using the power rule, the second using the chain rule and the last two come from the product rule. The derivatives of sin(x), ln(x) and exp(x) can be found in table of derivatives.

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 are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither.

In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3).

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.

Generalizations

For more details on this topic, see derivative (generalizations).

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'). Some people pronounce 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 and weak derivatives.

For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions.

References

Print

Spivak, Michael; Calculus (3rd edition, 1994) Publish or Perish Press. ISBN 0914098896.

Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480. Introduced by Martin Gardner.

Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.

Anton, Howard (1980). Calculus with analytical geometry.. New York:John Wiley and Sons. ISBN 0-471-03248-4.

Online books

Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin
Stroyan, K.D., A Brief Introduction to Infinitesimal Calculus, University of Iowa
Mauch, Sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus
Crowell, Benjamin, Calculus, Fullerton College, an online textbook
Garrett, Paul, Notes on First-Year Calculus
Hussain, Faraz, Understanding Calculus, an online textbook
Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus
Wikibook of Calculus

External links

WIMS Function Calculator makes online calculation of derivatives; this software also enables interactive exercises.

ADIFF online symbolic derivatives calculator.

DERIVATIVE

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