Chain Rule

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Chain Rule is the rule related to find out the derivatives. In this generally there are two functions. So this rule

determines the derivative where two functions are involved. This rule is very important part of differentiation.

Without chain rule the problem would be really complicated. This tool helps in easy calculation of

differentiation of two different functions. Its scope is not limited to this but it is so wide. It has so many other

applications as well.


The two examples will give clear indication about its worth.The examples are shown below.



Example 1:- Find the differentiation of   2 (2x+1) ^2


Solution:- Given function is 2 (2x+1) ^2 and this is a composite function.

 To find: - d/dx 2 (2x+1) ^2.

 Now d/dx 2 (2x+1) ^2 = 2 d/dx (2x+1) ^2 (because 2 is constant so comes out)

 So d/dx 2 (2x+1) ^2 = 2 x 2 (2x+1)^(2-1) d/dx (2x+1)

 4 (2x+1) x 2 = 8 (2x+1) = 16x +8

 So the differentiation of 2 (2x+1) ^2 is 16x +8.


Example 2:- Find the differentiation of   3 (3x+1) ^2.


Solution:- Given function is 3 (3x+1) ^2 and this is a composite function.

 To find: - d/dx 3 (3x+1) ^2.

 Now d/dx 3 (3x+1) ^2 = 3 d/dx (3x+1) ^2 (because 2 is constant so comes out)

 So d/dx 2 (2x+1) ^2 = 3 x 2 (3x+1)^2-1d/dx (3x+1)

 6 (3x+1) x 3 = 18 (3x+1) = 54x + 18

 So the differentiation of  3 (3x+1) ^2 is 54x +18.


 

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