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 (17361813).
An alternative notation for the derivative was previously introduced by the German mathematician Baron Wilhem Gottfried von Leibniz (16461716) who, independently and simultaneously with the Englishman Sir Isaac Newton (16421727), 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 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.
y  
f '(x)  =  lim  =  lim  
x>0  x  h>0  h 

=  f '(x) 

=  f^{(n)}(x) 

=  g'(u) 
Example. We have seen that if f(x) = x^{2} 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 = x^{2}, then

=  2x 

x^{2}_{ }  =  2x 
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.



A variation on the Leibniz notation that we will sometimes use is the following.
D_{x}  (f(x))  =  f '(x) 
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) = x^{2}, 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 
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William A. Bogley
Robby Robson