The Handy Math Answer Book

DIFFERENTIAL CALCULUS

What is differential calculus?

Differential calculus is the part of “the” calculus that deals with derivatives. It deals

with the study of the limit of a quotient, usually written as y/ x, as the denomina-

tor (x) approaches zero, with xand yas variables.

What is the derivative of a function?

One of the most important, core concepts in modern mathematics and calculus is the

derivative of a function—or a function derived from another function. A derivative is

also expressed as the limit of y/ x, also said as “the derivative of ywith respect to

x.” It is actually the rate of change (or slope on a graph) of the original function; the

derivative represents an infinitesimal change in the function with respect to the para-

meters contained within the function.

In particular, the process of finding the derivative of the function yf(x) is called

differentiation. The derivative is most frequently written as dy/ dx; it is also expressed

in various other ways, including f'(x) (said as the derivative of a function fwith respect

to x), y', Df(x), df(x), or Dxy. It is important to note that the differentials, written as dy

and dx,represent singular symbols and not the products of the two symbols. Not all

derivatives exist for all values of a function; the sharp corner of a graph, in which

there is no definite slope—and thus no derivative—is an example.

What is the standard notationfor the derivative?

The following represents the definition of the derivative of f(x) (note: in order for the

limit to exist, bothhlim$ 0 and hlim$ 0 must exist and be equal; thus, the function must be

continuous):

For f(x)’s derivative at point x 0 :

For f(x)’s derivative at xa:

Is there a formulafor the inverse of a derivative?

Yes. In this case, the derivative of the inverse function represents the inverse of y(x)—

or x(y):

ff() () ( ) ()

fa xa

xfa

h

lim lim ah fa

=xa - h 0

• =

+-

lak " "

ff() () ( ) ()

dx

df

xfx xx

xfx

h

xh fx

xx

lim

h

lim

== - 0

• =

+-

(^00) " (^00) "
aakkl^000

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