Tutorial on how to find derivatives of functions in calculus (Differentiation) involving the absolute value.
Example 1: Find the first derivative f '(x), if f is given by
f(x) = x  1
Solution to Example 1
Noting that  u  = √ ( u^{2} ), let u = x  1 so that f(x) may be written as
f(x) = y = √( u^{2} )
We now use the chain rule
f '(x) = (dy / du) (du / dx)
= (1/2) (2 u) / √ (u^{2}) (du / dx)
= u . u' /  u 
= u . 1 / √ (u^{2}) = (x  1) / x  1
We note that if x > 1, x  1 = x  1 and f '(x) = 1. If x < 1, x  1 = (x  1) and f '(x) = 1. f'(x) does not exist at x = 1.
Example 2: Find the first derivative of f given by
f(x) =  x + 2 +  x + 2
Solution to Example 2
f(x) is made up of the sum of two functions. Let u =  x + 2 so that
f'(x) = 1 + u u' / u = 1 + ( x + 2)(1) / x + 2
We note that if x < 2,  x + 2  =  x + 2 and f'(x) = 2 . If x > 2, f'(x) = 0. f'(x) does not exist at x = 2.
As an exercise, plot the graph of f and explain the results concerning f'(x) obtained above.
Exercises: Find the first derivatives of these functions
1. f(x) = 2x  5
2. f(x) = (x  2)^{2} + x  2
Answers to above exercises:
1. f'(x) = 2 (2x  5) / 2x  5
2. f(x) = 2 (x  2) + (x  2) / x  2
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