Differential calculus
Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called
differentiation.
Given a function and a point in the domain, the derivative at that
point is a way of encoding the small-scale behavior of the function near
that point. By finding the derivative of a function at every point in
its domain, it is possible to produce a new function, called the
derivative function or just the
derivative of the original function. In mathematical jargon, the derivative is a linear operator
which inputs a function and outputs a second function. This is more
abstract than many of the processes studied in elementary algebra, where
functions usually input a number and output another number. For
example, if the doubling function is given the input three, then it
outputs six, and if the squaring function is given the input three, then
it outputs nine. The derivative, however, can take the squaring
function as an input. This means that the derivative takes all the
information of the squaring function—such as that two is sent to four,
three is sent to nine, four is sent to sixteen, and so on—and uses this
information to produce another function. (The function it produces turns
out to be the doubling function.)
The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of
f is
f′, pronounced "f prime." For instance, if
f(
x) =
x2 is the squaring function, then
f′(
x) = 2
x is its derivative, the doubling function.
If the input of the function represents time, then the derivative represents change with respect to time. For example, if
f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of
f is how the position is changing in time, that is, it is the velocity of the ball.
If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as
y = mx + b, where
x is the independent variable,
y is the dependent variable,
b is the
y-intercept, and:
This gives an exact value for the slope of a straight line. If the
graph of the function is not a straight line, however, then the change
in
y divided by the change in
x varies. Derivatives give
an exact meaning to the notion of change in output with respect to
change in input. To be concrete, let
f be a function, and fix a point
a in the domain of
f. (
a,
f(
a)) is a point on the graph of the function. If
h is a number close to zero, then
a +
h is a number close to
a. Therefore (
a +
h,
f(
a +
h)) is close to (
a,
f(
a)). The slope between these two points is
This expression is called a
difference quotient. A line through two points on a curve is called a
secant line, so
m is the slope of the secant line between (
a,
f(
a)) and (
a +
h,
f(
a +
h)). The secant line is only an approximation to the behavior of the function at the point
a because it does not account for what happens between
a and
a +
h. It is not possible to discover the behavior at
a by setting
h to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the limit as
h tends to zero, meaning that it considers the behavior of
f for all small values of
h and extracts a consistent value for the case when
h equals zero:
Geometrically, the derivative is the slope of the tangent line to the graph of
f at
a.
The tangent line is a limit of secant lines just as the derivative is a
limit of difference quotients. For this reason, the derivative is
sometimes called the slope of the function
f.
Here is a particular example, the derivative of the squaring function at the input 3. Let
f(
x) =
x2 be the squaring function.
The slope of tangent line to the squaring function at the point (3,9)
is 6, that is to say, it is going up six times as fast as it is going
to the right. The limit process just described can be performed for any
point in the domain of the squaring function. This defines the
derivative function of the squaring function, or just the
derivative
of the squaring function for short. A similar computation to the one
above shows that the derivative of the squaring function is the doubling
function.
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