# What is the Product and Quotient Rule with Example

## Simple Definition of Derivative in Calculus

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

Derivative

### 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