How would you differentiate a function with a square root in the denominator?

d/dx 8/?(x-2)
(0*?(x-2) - 8/(2?(x-2)))/(?(x-2))²
(-4/?(x-2))/(x-2)
use keep, change, flip
-4/(x-2)^(3/2)

Well you can take it as...
d/dx8*(x-2)^-(1/2) and do the chain rule from there.
= 8(-1/2)(1)(x-2)^(-3/2)
=-4/(x-2)^(3/2)

Taking a square root of x is the same as raising x to the power of 0.5, or:

sqrt(x) = x^(1/2)

So, 8/sqrt(x-2) = 8/(x-2)^0.5 = 8*(x-2)^-0.5

Then, taking the derivative is easier.

Hint: d/dx x^a = a x^(a-1)

Derivative=8*-1/2*(x-2)^(-3/2)=-4/[(x-2)^(3/2)]

This is a little messy, especially with just the symbols available on a computer keyboard, but here's a general description, followed by some details:

The procedure is to
1. Write out the formula for f(x + h) and then stick the whole thing into the difference quotient.  
2.  "Simplify" by multiplying the top and bottom of the whole difference quotient by the product of the denominators in the numerator, which in this case means [sqrt(x + h - 2)] [sqrt(x - 2)].
3. Now at least you have no fractions inside the numerator or denominator.  Now rationalize the numerator by multiplying the top and bottom of the whole thing by the conjugate of the numerator (recall that the conjugate of a - b is a + b).
4.  Combine the like terms on the top, and you'll discover that h cancels out of the numerator and denominator.
5. Replace the h's in the denominator with zeros, since you're taking the limit as h approaches zero.
6.  Multiply the denominator out, which will provide some simplification for the radicals.

By the way, some books use "delta x" where I'm talking about h.  They're the same thing.


f(x+h) in this case is 8 / [sqrt(x+h-2)], and of course f(x) = 8 / [sqrt(x - 2)]

If you take the difference quotient and then multiply the numerator and denominator by the product
[sqrt(x + h - 2)] [sqrt(x - 2)], then you get:

{8 [sqrt(x - 2) - sqrt(x + h - 2) ] } / [h sqrt(x + h - 2) sqrt (x - 2)]

Now you rationalize the numerator by multiplying the numerator and denominator by [sqrt(x - 2) + sqrt (x + h - 2)].  This will give you a difference of squares on top, and so the result is

{8 [x - 2 - (x + h - 2) ] } / { [ h sqrt(x + h - 2) sqrt (x - 2) ] [ sqrt(x - 2) + sqrt(x + h - 2) ] }
or
(-8h) / { [ h sqrt(x + h - 2) sqrt (x - 2) ] [ sqrt(x - 2) + sqrt(x + h - 2) ] }
Now the h's cancel, so we get
-8 / { [ sqrt(x + h - 2) sqrt (x - 2) ] [ sqrt(x - 2) + sqrt(x + h - 2) ] }

if we now take the limit as h approaches 0, we get

-8 / { [x - 2] [ 2 sqrt (x - 2) ] }
or
-8 / [2 (x - 2)^(3/2)]
or
-4 / (x - 2)^(3/2)
or
-4 [ (x - 2)^(-3/2) ]

(and if you already know the Chain rule, then you can see that this is the answer you're supposed to get.  If you haven't learned the chain rule yet, then you can look forward to it with great anticipation, because once you learn it you'll never have to do this again!)