Question

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

Answer 1

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))`