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

1 Answer
Dec 28, 2015

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

Explanation:

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))