shreyadas

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Business Mathematical

 Introduction
In Economics and Commerce, the concept of rate is very commonly used and is important.
 Example
We are interested in finding out how fast the price of a commodity or a share is changing with time.We are interesting in knowing how fast (or slow) the supply quantity of a product in the market changes either change in its price. We are interested in finding out the rate at which the total revenue changes with respect to a change in quantity sold.
In general, if y=f(x) is a function involving a dependent variable y which depends on an independent variable x, then we may be interested in finding out the rate of change of y values with repect to a change in x values. This rate is called the “DERIVATIVE”. It gives the instantaneous rate of the curve y=f(x) at the instant or at the point x. It is denoted by dy/dx or by f’(x).






 RULES OF DERIVATION
There are some simple rules of derivative which help in finding derivatives of some more complicated functions. If one can find the derivative of a function at each point x, we call it a differentiable function. If the function is a sum, difference, scalar multiple, product or a quotient of differentiable functions, then we can use therules of differentiation stated below.
1. Derivatives of sum
If u and v are differentiable fuctions of x and y=u+v then
dy/dx=du/dx+dv/dx
this rule can be remembered as:
Derivative of the sum of two functions is equal to the sum of their derivatives.
Corollary:-
This rule can be extended further as,
If y= =u+v+w, when,
dy/dx=du/dx+dv/dx+dw/dx¬¬


Example
If y=a^x+x^a+e^x, then find, dy/dx.

Solution

Dy/dx=d/dx (a^x) + d/dx (x^a) + d/dx (e^x)
=a^x log a + ax^a-1 + e^x


2. Derivatives of difference
If u and v are differentiable functions of x and y = u – v, then,
dy/dx= du/dx – dv/dx
This rule can be remembered as :
Derivative of the difference of two functions is equal to the difference of their derivatives.


3. Derivatives of a product
If u and v are differentiable functions of x and y = uv, then
dy/dx = u. dv/dx + v. du/dx

This rule can be remembered as follows:
Derivative of the product of two functions
=(first function) x (derivative of the second function) + (second function) x (derivative of the first
function).

Corollary
If u, v, w are differentiable function of x and y = uvw then,
dy/dx =uv. dw/dx + uw. dv/dx + vw. du/dx
Example
If y = x^3 . e^x; Find dy/dx.

Solution
y = x^3 . e^x. Hence,
Dy/dx = x^3. d/dx (e^x) + e^x. D/dx (x^3)
=x^3 (e^x) + e^x (3x^2)
=x^2 e^x (x+3)


4. Derivative of quotient
If u and v are differentiable function of x and y = u/v where v = 0, then
Dy/dx = v.du/dx – u. dv/dx/v^2
 
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