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 realvalued 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
variablethe 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)

sec^{2}(x)

cot(x)

− csc^{2}(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
Definition
The logarithm of a number y with respect to base b is the exponent to which b has to be raised to yield y. In other words, the
logarithm of y to base b is the solution x of the equation
The logarithm is denoted "log_{b}(y)"
(pronounced as "the logarithm of y to base b" or "the
"baseb logarithm of y"). In the equation log_{b}(y)
= x, the value x, is the answer to the
question "To what power must b be raised, in order to yield y?"For the logarithm to be
defined, the base b must be a positive real number not equal to 1 and y must be a positive number
Derivative
Analytic
properties of functions pass to their inverses. Thus, as f(x) = b^{x} is a continuous and differentiable function, so
is log_{b}(y). Roughly, a continuous function is
differentiable if its graph has no sharp "corners". Moreover, as the derivative of f(x) evaluates to ln (b)
bx by the properties of the exponential function, the chain rule implies that the derivative
of log_{b}(x) is given by
That is, the slope of the tangent touching the graph of the baseb logarithm at the point (x, log_{b}(x)) equals 1/(x ln(b)). In
particular, the derivative of ln(x) is 1/x, which implies that
the antiderivative of 1/x is ln(x)
+ C. The derivative with a generalised functional argument f(x) is
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Logarithm#Derivative_and_antiderivative
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