Derivative of Absolute Value

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To understand the derivative of absolute value, first we need to understand the meaning of absolute value. Absolute value is defined as the non negative value of a number. Suppose y is any positive number then absolute value is represented by the |y|. Even if y is any negative number then also |y| = y. The absolute value of any number whether number is positive or negative, is always positive. Derivative of absolute value help us to find the derivative of the absolute value of any function.
The formula of derivative of absolute value is as follows:-

Derivative, d/dx |x| = (x. dy/dx) / |x|, x shall not be equal to zero     

This can be more clarified by the following below mentioned examples:- 

Question 1: Find out the derivative of the function y = |x-2|

Solution: Given y = |x-2|

Now dy/ dx = d |x-2| / dx|x-2|

So, dy/dx = ((x-2). (d(x-2)) / dx) / |x-2|

Hence dy/dx = (x-2). 1 / |x-2|

Therefore, dy/dx = (x-2) / |x-2|

Since denominator becomes 0 at x =2

So derivative of function dy/dx does not exist at x = 2.
Question 2: Find out the differentiation of y = |x^2| and find the value of dy/dx at x = 2.

Solution: Given, y = |x^2|

Therefore by definition, dy /dx = (x^2. d (x^2) /dx) / |x^2|

So dy/dx = (x^2. 2x) / |x^2|

Now at x = 2, dy/dx = (2^2.  2(2)) / |2^2|

Therefore at x = 2, dy/dx = (4.4)/ 4 = 4

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