NOTATION: LAGRANGE AND LEIBNIZ


There are a variety of notational conventions for the derivative in widespread use. We have already introduced the "prime" notation for the derivative, in which the derivative of f is denoted by f '. The prime notation was introduced by the French mathematician Comte Joseph Louis Lagrange (1736-1813).

An alternative notation for the derivative was previously introduced by the German mathematician Baron Wilhem Gottfried von Leibniz (1646-1716) who, independently and simultaneously with the Englishman Sir Isaac Newton (1642-1727), was credited with the development of calculus. Both of these notational schemes are in widespread use today, principally because they have complementary advantages and disadvantages.

The Leibniz notation derives from the use of the capital letter Delta to indicate the change in a variable quantity. If the function f is differentiable at x and we set y = f(x), then the derivative f '(x) is defined as follows.

Deltay f(x + h) - f(x)
f '(x)=lim
=lim
Deltax-->0Deltaxh-->0h

If we squint our eyes and look at the middle term of this equation, our focus becomes distorted and we lose sight of the "limit" symbol. Eventually, each of the Delta's begins to look like the letter "d", and we find ourself writing the following.
dy

dx
=  f '(x)

For higher order derivatives, we alter this notation as follows.

dny

dxn
=  f(n)(x)

The notation adjusts for changes in the independent and dependent variables. For instance, if z = g(u), then we use Leibniz notation for the derivative as follows.

dz

du
=  g'(u)

Advantages of this Leibniz notation for the derivative become apparent in the following examples.


Example. We have seen that if f(x) = x2 for all x in R, then f '(x) = 2x for all x in R. In the Leibniz notation, this assertion is stated as follows. If y = x2, then

dy

dx
=  2x

Even more economically, the symbol d/dx is often used in isolation to indicate the process of differentiation.

d

dx
x2  =  2x

The advantage of the Leibniz notation is that we can indicate the derivative of a function, such as the squaring function, without having to introduce a separate name for the function (such as f). In many instances, this economy of expression is extremely helpful.


Old Examples: New Outlook. Here are some other differentiation formulas that we have already verified using the definition of the derivative as the limit of difference quotients. We have already recorded these formulas in the prime notation. Here we use a variety of independent variables in order to illustrate the flexibility of the notation.

d

dz
sqrt(z) =
1

2sqrt(z)
d

dt
2t  =  2
d

dw
=  0

For instance, the last of these is a succinct statement of the fact that if f is the constant function given by f(w) = 2 for all w in R, then f '(w) = 0 for all w in R.


A variation on the Leibniz notation that we will sometimes use is the following.

Dx (f(x)) =  f '(x)

Thus, the symbol Dx is a synonym for d/dx.


For all of its economy and power of expression, the Leibniz notation is at a true disadvantage if one is interested in a particular value  of the derivative. If a function f is differentiable at a, then the value of the derivative at a is denoted simply by f '(a). For example, if f(x) = x2, then f '(7) = 2(7) = 14.

The Leibniz notation does not accomodate evaluation at specific values without some additional decoration. When necessary, this is done as follows.

dy |

= f '(a)
dx x = a

In practice, we will use whichever notation is most convenient, and so both the Leibniz and the prime notation will be used throughout our studies. You will need to become accustomed to using both conventions.


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William A. Bogley
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