In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes.
Loosely speaking, a derivative can be thought of as how much one quantity is
changing in response to changes in some other quantity; for example, the
derivative of the position of a moving object with respect to time is the
object's instantaneous velocity (conversely, integrating a car's velocity over time yields the
distance traveled).
The derivative
of a function at a chosen input value describes the best linear approximation of the function near that input
value. For a real-valued function of a single real variable, the
derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher
dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function.
The process of
finding a derivative is called differentiation.
Differentiation is a method to compute
the rate at which a dependent output y changes with respect to the change in
the independent input x.
This rate of change is called the derivative of y with respect to x. In more precise language,
the dependence of y upon x means that y is a function of x.
This functional relationship is often denoted y = ƒ(x),
where ƒ denotes the function. If x and y are real
numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.
The simplest case is when y is a linear
function of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x)
= m x + b,
for real numbers m and b,
and the slope m is given by
where the
symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for
"change in."
http://en.wikipedia.org/wiki/Derivative
Differentiation of
trigonometric functions
The differentiation of
trigonometric functions is the mathematical process of finding the rate at which a trigonometric function changes with respect to a
variable--the derivative of the trigonometric function. Commonplace trigonometric
functions include sin(x), cos(x) and tan(x). For example,
in differentiating f(x) = sin(x), one is calculating a function f ′(x) which
computes the rate of change of sin(x) at a particular point a. The value of the rate of
change at a is thus given by f ′(a). Knowledge of differentiation from first
principles is required, along with competence in the use of trigonometric identities and limits. All functions
involve the arbitrary variable x, with all differentiation performed with respect to x.
|
Function
|
Derivative
|
|
sin(x)
|
cos(x)
|
|
cos(x)
|
− sin(x)
|
|
tan(x)
|
sec2(x)
|
|
cot(x)
|
− csc2(x)
|
|
sec(x)
|
sec(x)tan(x)
|
|
csc(x)
|
− csc(x)cot(x)
|
|
arcsin(x)
|
|
|
arccos(x)
|
|
|
arctan(x)
|
|
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions
Implicit and explicit functions
In mathematics, an implicit function is a function in which the dependent variable has not been given
"explicitly" in terms of the independent variable. To give a function f explicitly is
to provide a prescription for determining the output value of
the function y in terms of the input value x:
y = f(x).
By contrast, the function is implicit if
the value of y is obtained from x by solving an
equation of the form:
R (x,
y) = 0.
That is, it is defined as the level
set of a function in two variables: one variable or the other may determine the
other, but one is not given an explicit formula
for one in terms of the other.
Implicit
differentiation
In calculus, a
method called implicit
differentiation makes use of
the chain rule to differentiate implicitly defined
functions.
As explained in the introduction, y can be given as a function of x implicitly rather than explicitly.
When we have an equation R(x, y) = 0,
we may be able to solve it for y and then differentiate. However,
sometimes it is simpler to differentiate R(x, y)
with respect to x and then solve for dy/dx.
Example
Consider for example
This function normally can be manipulated by using algebra to change this equation to an explicit function:
Differentiation then gives 
.
Alternatively, one can differentiate the equation:
Solving for
:
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Implicit_and_explicit_functions
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