## Define Homogeneous Function with Example

A function f defined by

u\;=\;f\;(x,\;y,\;z,\;…)

of any number of variables is said to be homogeneous of degree n in these variables if multiplication of these variables by any number t(\neq0) result in the multiplication of the function by t^n, i.e.,

f\;(tx,\;ty,\;tz,\;…) = t^n f\;(x,\;y,\;z,\;…) (1)

provided that (tx,\;ty,\;tz,\;…) is in the domain of f.

Taking t=\frac1x,(x\neq0) the equation (1) becomes

• f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;=\;\frac1{x^n}f(x,\;y,\;z,\;…)\;
• \;f(x,\;y,\;z,\;…)\;\;=x^n\;f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;
• =x^ng\left(\frac yx,\;\frac zx\;…\right)

## Simple Definition of Derivative in Calculus

Derivative is the rate of change of a function f(x) with respect to any variable.

### Derivative of a Function Examples

Let f be a real valued function Continuous in the interval (x,x1) Df the domain of (f) , then
different quotient= f(x1) – f(x)/x1 – x
represents the average rate of change in the value of f with respect to change x1 – x in the values of independent variable x.

### Uses of Derivatives in Economics

• we mostly use derivative to determine minimum and maximum values of a particular function f(x).
• Derivatives are also used in many engineering and science problems.
• Specially modelling the behavior of any moving object.

### How Can We Use Derivatives in our Daily Life?

• Derivative is mostly used in calculating profit and loss in business by using different graphs.
• ⇒ it is also used to determine speed or any distance covered i.e, miles per hour or kilometer(km) per hour.
• ⇒ Derivatives are most commonly used in physics to drive different equations.

### How to Find Derivative of any Function f(x)?

Basically it is obtained by moving the power in the start and decreasing the power by 1.

### Example for Derivative of any Function f(x)

Derivative of x⁴ is 4x³. Note that power is in the start and power is decreased by 1

Derivative of x12 is 12x12-1 = 12x11

#### Some more examples for Derivative of Function

\begin{array}{l}4x^3+\;x^2\;\\=\;\frac d{dx}\left(4x\right)^3\;+\;\;\frac d{dx}\left(x\right)^2\\=3\left(4x\right)^{3-1}\;+\;2\left(x\right)^{2-1}\\=12x^2\;+\;2x\\or\;2x(6x\;+\;1)\end{array}


### Derivative Notation and Mathematician Names

Now, we write in the table the notations for the derivatives of y=f(x) used by different mathematicians.

### Basic Rules of Differentiation with Examples

⇒ Derivative of a constant is always zero i.e., derivative of (x² + 1) is

\begin{array}{l}x^2+\;1\;\\=\;\frac d{dx}\left(x\right)^2\;+\;\;\frac d{dx}1\\=2\left(x\right)^{2-1}\;+\;0\\=2x\\\end{array}

⇒Derivative of d/dx(x) =1
In mathematics a,b,c are mostly considered constants. So, derivative of a,b,c are also zero(0).

Derivative of a sum or a difference of any function
in simple manners:
if f and g are differentiable at x then,
[f(x) + g(x)]’
= f’(x) – g’(x)
that is,
=d/dx[f(x) + g(x)]
= d/dx[f(x) + d/dx [g(x)]

also,
=[f(x) – g(x)]’= f’(x) – g’(x)
that is,
=d/dx [f(x) – g(x)]
=d/dx [f(x)] -d/dx[g(x)]

## What is the Product and Quotient Rule with Example

Product and Quotient rule with formula are as follow;

### What is Product Rule in Differentiation?

if f and g are differentiable at x , then fg is also differentiable at x and
[f(x)g(x)]’ = f’(x) g(x) + f(x) g’

that is,

d/dx[f(x) g(x)] = [d/dx[f(x)] g(x) + f(x) [d/dx[g(x)]

### What is Quotient Rule in Differentiation

if f and g are differentiable at x and g≠0,
for any x∈ D(g) then f/g is differentiable at x and [f(x)/g(x)]’ = f’(x) g(x) – f(x) g’(x)/[g(x)]²

that is,

d/dx[f(x)/g(x)]’
= [d/dx[f(x)] g(x) – f(x)[d/dx[g(x)]/[g(x)]²

## what is the product rule for derivatives?

Definition of Product rule in easy words:

The product rule tells us how to differentiate or solve of two or three function with respect to derivative function.

Here, let us suppose that two functions are g and f. Now in product rule, first(f) like it and derivative of a second(g) than the second(g) as it and derivative of first(f)

In the Product Rule, the derivative of a made from features is the first function times the derivative of the second function plus the second fun instances the by-product of the primary feature.

## Quotient Rule Examples with Solutions

Some important, basic, and easy examples are as follows:

But before examples, we discuss what is Quotient Rule in Calculus is.

### Definition of Quotient Rule in Math?

In Mathematics, While you divide two(2) numbers the answer is referred to as the Quotient.

or

The quotient rule is a technique for differentiating issues in which one feature is divided by using any other. If differentiable functions, f(x) and g(x), exist, then their quotient is also differentiable (i. E., the by-product of the quotient of those two capabilities also exists)