Help with derivatives

**mrge** · February 8th 2010, 10:03 PM

Here is the problem
Given: Given f(x) = ( sqrt ( x ) - 3) / (sqrt ( x ) + 3 ).

Find the derivative. I seem to be very weak at my algebra, im not sure what to do after this:

(sqrt (x) +3 (1/2 x ^-1/2 ) - sqrt (x) - 3 (1/2 x ^-1/2 ) / (sqrt (x) +3)^2 This comes from the quotient rule. Can anyone give me some tips to simplify the equation?

**Prove It** · February 8th 2010, 10:33 PM

Originally Posted by mrge

Here is the problem
Given: Given f(x) = ( sqrt ( x ) - 3) / (sqrt ( x ) + 3 ).

Find the derivative. I seem to be very weak at my algebra, im not sure what to do after this:

(sqrt (x) +3 (1/2 x ^-1/2 ) - sqrt (x) - 3 (1/2 x ^-1/2 ) / (sqrt (x) +3)^2 This comes from the quotient rule. Can anyone give me some tips to simplify the equation?

You can either use the quotient rule or simplify the function first.

Quotient rule:

$f(x) = \frac{\sqrt{x} - 3}{\sqrt{x} + 3}$

$= \frac{x^{\frac{1}{2}} - 3}{x^{\frac{1}{2}} + 3}$ .

So $f'(x) = \frac{(x^{\frac{1}{2}} + 3)(x^{\frac{1}{2}} - 3)' - (x^{\frac{1}{2}} - 3)(x^{\frac{1}{2}} + 3)'}{(x^{\frac{1}{2}} + 3)^2}$

$= \frac{\frac{1}{2}x^{-\frac{1}{2}}(x^{\frac{1}{2}} + 3) - \frac{1}{2}x^{-\frac{1}{2}}(x^{\frac{1}{2}} - 3)}{(x^{\frac{1}{2}} + 3)^2}$

$= \frac{\frac{1}{2}x^{-\frac{1}{2}}(x^{\frac{1}{2}} + 3 - x^{\frac{1}{2}} + 3)}{(x^{\frac{1}{2}} + 3)^2}$

$= \frac{3x^{-\frac{1}{2}}}{(x^{\frac{1}{2}} + 3)^2}$

$= \frac{3}{\sqrt{x}(\sqrt{x} + 3)^2}$ .

**mrge** · February 8th 2010, 11:10 PM

oh i see.. thanks!!

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