Using the limit definition, how do you differentiate #f(x) = x^(1/2) #?

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Answer 1 · 2015-12-28T18:38:27

The crucial step is #((x+h)^(1/2) - x^(1/2))/h = h/(h((x+h)^(1/2) + x^(1/2)))#

Here is the algebra of the crucial step of "rationalizing" the numerator:

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

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

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

Using this algebra, we get:

#lim_(hrarr0) ((x+h)^(1/2) - x^(1/2))/h = lim_(hrarr0)h/(h((x+h)^(1/2) + x^(1/2)))#

# = lim_(hrarr0) 1/((x+h)^(1/2) + x^(1/2))#

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