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 Question from Eric, a student: I have an problem figuring out the derivative of the negative square root of x i.e. x^-(1/2) using the first principle. Could someone please show me? Thanks in advance!

Hi Eric,

Let me use $f(x) = x^{-1/2} = \large \frac{1}{x^{1/2}}.$ You need to work with

$f(x + h) - f(x) = \frac{1}{(x + h)^{1/2}} - \frac{1}{x^{1/2}}.$

Using the common denominator of $x^{1/2} \times (x + h)^{1/2}$ and simplifying I get

$f(x + h) - f(x) = \frac{x^{1/2} - (x + h)^{1/2}}{x^{1/2} \times (x + h)^{1/2}}.$

At this point multiply the numerator and denominator by $x^{1/2} + (x + h)^{1/2},$ use the fact that you have a difference of squares in the numerator and simplify.

See if you can complete the derivation from here. Write back if you need more assistance,
Harley

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