Differentiation, Algebra of Derivatives, Derivatives of Sum and Difference of Functions, Derivative of Product of Functions

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Algebra of Derivatives

  • Many functions arise as combinations of other functions. The combination could be sum, difference, product or quotient of functions. We also come across situations where a given function can be expressed as a function of a function.

  • In order to make derivative as an effective tool in such cases, we need to establish rules for finding derivatives of sum, difference, product, quotient and function of a function. These, in turn, will enable one to find derivatives of polynomials and algebraic (including rational) functions.

Derivatives of Sum and Difference of Functions:

If and are both derivable functions and , then what is ?

Here

Let be the increment in and be the corresponding increment in.

Hence

Or

Thus we see that the derivative of sum of two functions is sum of their derivatives.

This is called the Sum Rule.

This sum rule can easily give us the difference rule as well, because

If

Then

I.e. the derivative of difference of two functions is the difference of their derivatives.

This is called Difference Rule.

Sum Rule:

Difference Rule:

Example:

Find the derivative of each of the following functions:

(i)

(ii)

Solution:

(i) We have,

Derivative of Product of Functions:

You are all familiar with the four fundamental operations of Arithmetic: addition, subtraction, multiplication and division. Having dealt with the sum and the difference rules, we now consider the derivative of product of two functions.

In general, if and are two functions of then the derivative of their product is defined by

Which is read as derivative of product of two functions is equal to

This is called the Product Rule.

Example:

Find , if

Solution:

Here is a product of two functions.

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