Table of Contents

## What is Differentiation?

Differentiation is the process of mathematically finding the derivative of a function. The derivative is a measure of how a function changes as its input changes. It can be used to answer questions such as how fast a particular variable is growing or shrinking, or how a particular curve is changing.

## What is Derivative?

A derivative is a financial contract that derives its value from an underlying asset. The most common underlying assets are stocks, bonds, commodities, and currencies. Derivatives can be used for hedging (protecting against losses) or speculating (gambling on price changes).

## What are the Differentiation Rules?

There are three types of differentiation rules: product, quotient, and chain.

The product rule states that the derivative of the product of two functions is the product of their derivatives.

The quotient rule states that the derivative of the quotient of two functions is the quotient of their derivatives, divided by the the denominator of the quotient.

The chain rule states that the derivative of the composition of two functions is the derivative of the first function, multiplied by the derivative of the second function, taken at the point where the first function is being applied.

## We are Going to Discuss Product Rule in Detail

The product rule is a mathematical theorem that states that the derivative of the product of two functions is the product of their derivatives. In symbols,

This theorem is useful in calculating derivatives of composite functions.

The product rule can be derived from the definition of the derivative of a function. Let

be a function, and let

and

be two differentiable functions. Then, the derivative of

is

This follows from the definition of the derivative of a function, which is given by

The derivative of a product is the product of the derivatives, so

The proof of the product rule is a little more involved. First, note that

This follows from the chain rule, which states that the derivative of the composite of two functions is the derivative of the first function multiplied by the derivative of the second function.

Now, let

be a function, and let

and

be two differentiable functions. Then, the derivative of

is

This follows from the product rule, since the derivative of a product is the product of the derivatives.

## A Few Differentiation Formulas and Examples Have Been Listed Below:

-Distance Formula: (Distance) = ( Rate x Time)

-Area Formula: (Area) = (Base x Height)

-Volume Formula: (Volume) = (Length x Width x Height)

## Product Rule for the Logarithms to Write an Equivalent Sum of the Logarithms

The logarithm of a sum is the sum of the logarithms.