Math CentralQuandaries & Queries


Question from Carla, a student:

Find the derivative using the limit process of
f(x) = (x+1)^1/2

Hi Carla,

The derivative of $f(x)$ is the limit, as $h$ approaches zero of the difference quotient

\[\frac{f(x + h) - f(x)}{h}.\]

For your function this difference quotient is

\[\frac{(x + h + 1)^{1/2} - (x + 1)^{1/2}}{h}.\]

If you multiply the numerator and denominator by $(x + h + 1)^{1/2} + (x + 1)^{1/2}$ and simplify you will arrive at an expression that can be evaluated as $h$ approaches zero.

I hope this helps,

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