How do you find the derivative of #sqrt(x^2)#?

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Answer 1 · 2016-03-22T18:26:24

#d/dx(sqrt(x^2)) = x/sqrt(x^2) = {(1,"if",x > 0),(-1,"if",x<0):}#

#f(x) = sqrt(x^2) = (x^2)^(1/2)#

Using the chain rule we get:

#f'(x) = 1/2 (x^2)^(-1/2) d/dx(x^2)#

# = 1/2 (x^2)^(-1/2) * 2x#

# = x(x^2)^(-1/2)#

# = x/(x^2)^(1/2)#

# = x/sqrt(x^2)#.

This can also be written as a piecewise function.

#f(x) = sqrt(x^2) = absx = {(x,"if",x >= 0),(-x,"if",x<0):}#

So,

#f'(x) = {(1,"if",x > 0),(-1,"if",x<0):}#