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Explain Differentiation

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									                      Explain Differentiation
Explain Differentiation

Differentiation is another important part of the calculus besides Integration. Derivative
of a function can be explained as the rate of change of the function. Differential
equations are used in the minima and maxima problems.

This is the technique to find out the minima and maxima on the graph. Another way to
explain differentiation is its applications. The use of differentiation is in the increasing
or decreasing functions.

When a function f(x) increases with the increase in the input x then the function is
called as increasing function and when the value of the function decreases with the
input value then the function is called the decreasing function.

A function can be increased at some values or decrease on some other values. The
value where the function gets changed is called the turning point and a turning point
is a part of stationary point.


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Stationary point on the graph is the point where the gradient of the function is zero. At
maximum points the gradient of the function is positive just before the maxima and at
minimum points the gradient is negative, zero and then rise to position.

The derivative of any function tells you that how the output of the function changes,
as you make change in the input. In geometrical language, we define differentiation
as if, we draw the graph of the function then, the point on the slop of tangent line is
the derivative.

According to the definition of differentiation, we can say that it is a process of finding
derivatives. The opposite of differentiation, is called Integration. If we take the a
general example, let 'x' is any distance and 't' is the time than, differentiation of 'x' with
respect to 't' will gives velocity.

How to proof of the derivative of a constant? Before going to study of the derivative of
a constant, we should know about the derivative. Derivative of a function can be find
out very easily the important thing that we must have is knowledge about the
derivative. Derivatives have wide application in mathematics, as we can use them in
trigonometry, calculus and algebra.

With the help of derivative we can derive many equations which are useful in
mathematics. If you are asked to find out the derivative of a constant, then the answer
of this question is 0.

The reason for this is that, if we have to find derivative with respect to x then any
variable that not containing x is a constant for that. That is why the derivative of a
constant is 0.


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The concept of a derivative is invented from the problem to finding a tangent of a
curve because tangent of curve defines rate of change. So, let’s discuss history of
differentiation:

The first scientist who resembled the modern method to determining tangent of curve
is Giles Persone de Roberval during 1630’s and 1640’s. Roberval determine the
constituent motion vectors at a point.

Roberval define motion vectors, for a parabola and gives method to find tangent of
curve on parabola.

At same time, Pierre de Fermat used the notion of Maxima and Minima to finding
tangent of curve. Fermat dividing one line segment into two segment such that
product of two line segment was maximum.

Like a is a line segment and when we divide a into x and a-x, then product of x and a-
x gives maxima. Fermat give, definition of differentiation:




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