Engineering Handbook/Mathematics/Derivative

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Derivative[edit]

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" (or infinitesimal) increments of x and y, just as Δx and Δy represent finite increments of x and y.[1] For y as a function of x, or

y=f(x) \,,

the derivative of y with respect to x, which later came to be viewed as

\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x)-f(x)}{(x + \Delta x)-x},

was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or

\frac{dy}{dx}=f'(x),

where the right hand side is Lagrange's notation for the derivative of f at x. From the point of view of modern infinitesimal theory, \Delta x is an infinitesimal x-increment, \Delta y is the corresponding y-increment, and the derivative is the standard part of the infinitesimal ratio:

f'(x)={\rm st}\Bigg( \frac{\Delta y}{\Delta x} \Bigg).

Equivalently, f'(x) is the real number adequal to the infinitesimal ratio \frac{\Delta x}{\Delta y}. Then one sets dx=\Delta x, dy = f'(x) dx\,, so that by definition, f'(x)\, is the ratio of dy by dx.

Similarly, although mathematicians sometimes now view an integral

\int f(x)\,dx

as a limit

\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,

where Δx is an interval containing xi, Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities f(x) dx. From the modern viewpoint, it is more correct to view the integral as the standard part of, or adequal to, an infinite sum of such quantities.

History[edit]

The Newton-Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the \int character. He based the character on the Latin word summa ("sum"), which he wrote ſumma with the elongated s commonly used in Germany at the time. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum of June 1686,[2] but he had been using it in private manuscripts at least since 1675.[3]

In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Cauchy, Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above. Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard f(x) as measured in meters per second, and dx in seconds, so that f(x) dx is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis.

In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson introduced ways of treating infinitesimals both literally and logically rigorously, as a foundation for both calculus and analysis. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based to Robinson's approach.

Leibniz's notation for differentiation[edit]

In Leibniz's notation for differentiation, the derivative of the function f(x) is written:

\frac{d\bigl(f(x)\bigr)}{dx}\,.

If we have a variable representing a function, for example if we set

y=f(x) \,,

then we can write the derivative as:

\frac{dy}{dx}\,.

Using Lagrange's notation, we can write:

\frac{d\bigl(f(x)\bigr)}{dx} = f'(x)\,.

Using Newton's notation, we can write:

\frac{dx}{dt} = \dot{x}\,.

For higher derivatives, we express them as follows:

\frac{d^n\bigl(f(x)\bigr)}{dx^n}\text{ or }\frac{d^ny}{dx^n}

denotes the nth derivative of ƒ(x) or y respectively. Historically, this came from the fact that, for example, the third 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 \bigl(f(x)\bigr) =
\frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)\,.

Now drop the parentheses and we have:

\frac{d^3}{dx^3}\bigl(f(x)\bigr)\ \mbox{or}\ \frac{d^3y}{dx^3}\,.

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear to cancel:

\frac{dy}{dx} = \frac{dy}{du_1} \cdot \frac{du_1}{du_2} \cdot \frac{du_2}{du_3}\cdots \frac{du_n}{dx}\,,

etc., and:

\int y \, dx = \int y \frac{dx}{du} \, du.


Notations for differentiation[edit]

Main page: Notation for differentiation

Leibniz's notation[edit]

Main page: Leibniz's notation

The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still commonly used when the equation y = ƒ(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by

\frac{dy}{dx},\quad\frac{d f}{dx}(x),\;\;\mathrm{or}\;\; \frac{d}{dx}f(x),

and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation

\frac{d^ny}{dx^n},
\quad\frac{d^n f}{dx^n}(x),
\;\;\mathrm{or}\;\;
\frac{d^n}{dx^n}f(x)

for the nth derivative of y = ƒ(x) (with respect to x). These are abbreviations for multiple applications of the derivative operator. For example,

\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right).

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

\left.\frac{dy}{dx}\right|_{x=a} = \frac{dy}{dx}(a).

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:[4]

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

Lagrange's notation[edit]

Sometimes referred to as prime notation,[5] one of the most common modern notations for differentiation is due to Joseph-Louis Lagrange and uses the prime mark, so that the derivative of a function ƒ(x) is denoted ƒ′(x) or simply ƒ′. Similarly, the second and third derivatives are denoted

(f')'=f''\,   and   (f'')'=f'''.\,

To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses:

f^{\mathrm{iv}}\,\!   or   f^{(4)}.\,\!

The latter notation generalizes to yield the notation ƒ (n) for the nth derivative of ƒ — this notation is most useful when we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can become cumbersome.

Newton's notation[edit]

Main page: Newton's notation

Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If y = ƒ(t), then

\dot{y}   and   \ddot{y}

denote, respectively, the first and second derivatives of y with respect to t. This notation is used exclusively for time derivatives, meaning that the independent variable of the function represents time. It is very common in physics and in mathematical disciplines connected with physics such as differential equations. While the notation becomes unmanageable for high-order derivatives, in practice only very few derivatives are needed.

Euler's notation[edit]

Euler's notation uses a differential operator D, which is applied to a function ƒ to give the first derivative Df. The second derivative is denoted D2ƒ, and the nth derivative is denoted Dnƒ.

If y = ƒ(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written

D_x y\,   or   D_x f(x)\,,

although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression.

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

Table of Derivatives
{d \over dx} c = 0
{d \over dx} x = 1
{d \over dx} cx = c
{d \over dx} |x| = {x \over |x|} = \sgn x,\qquad x \ne 0
{d \over dx} x^c = cx^{c-1} where both xc and cxc-1 are defined.
{d \over dx} \left({1 \over x}\right) = {d \over dx} \left(x^{-1}\right) = -x^{-2} = -{1 \over x^2}
{d \over dx} \left({1 \over x^c}\right) = {d \over dx} \left(x^{-c}\right) = -{c \over x^{c+1}}
{d \over dx} \sqrt{x} = {d \over dx} x^{1\over 2} = {1 \over 2} x^{-{1\over 2}}  = {1 \over 2 \sqrt{x}} x > 0
{d \over dx} c^x = {c^x \ln c} c > 0</math>
{d \over dx} e^x = e^x
{d \over dx} \log_c x = {1 \over x \ln c} c > 0,  c \ne 1
{d \over dx} \ln x = {1 \over x}
{d \over dx} \sin x = \cos x
{d \over dx} \cos x = -\sin x
{d \over dx} \tan x = \sec^2 x
{d \over dx} \sec x = \tan x \sec x
{d \over dx} \cot x = -\csc^2 x
{d \over dx} \csc x = -\csc x \cot x
{d \over dx} \arcsin x = { 1 \over \sqrt{1 - x^2}}
{d \over dx} \arccos x = {-1 \over \sqrt{1 - x^2}}
{d \over dx} \arctan x = { 1 \over 1 + x^2}
{d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}
{d \over dx} \arccot x = {-1 \over 1 + x^2}
{d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}
{d \over dx} \sinh x = \cosh x
{d \over dx} \cosh x = \sinh x
{d \over dx} \tanh x = \mbox{sech}^2 x
{d \over dx} \mbox{sech} x = - \tanh x \mbox{sech} x
{d \over dx} \mbox{coth} x = - \mbox{csch}^2 x
{d \over dx} \mbox{csch} x = - \mbox{coth} x \mbox{csch} x
{d \over dx} \mbox{arcsinh} x = { 1 \over \sqrt{x^2 + 1}}
{d \over dx} \mbox{arccosh} x = { 1 \over \sqrt{x^2 - 1}}
{d \over dx} \mbox{arctanh} x = { 1 \over 1 - x^2}
{d \over dx} \mbox{arcsech} x = { 1 \over x\sqrt{1 - x^2}}
{d \over dx} \mbox{arccoth} x = { 1 \over 1 - x^2}
{d \over dx} \mbox{arccsch} x = {-1 \over |x|\sqrt{1 + x^2}}

References[edit]

  1. Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5. 
  2. Mathematics and its History, John Stillwell, Springer 1989, p. 110
  3. Early Mathematical Manuscripts of Leibniz, J. M. Child, Open Court Publishing Co., 1920, pp. 73–74, 80.
  4. In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors. Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx. Others define "dx" as an independent variable, and define du by du = dx•ƒ′ (x). In non-standard analysis du is defined as an infinitesimal. It is also interpreted as the exterior derivative du of a function u. See differential (infinitesimal) for further information.
  5. [1]