# Math Help - Help with derivatives

1. ## Help with derivatives

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?

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

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

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

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

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

3. oh i see.. thanks!!