"Messy" Differentiation

[sofiaahmed]

Hi I am doing a problem that is getting a little messy, which is making it
hard to solve. The question is this: find the derivative of the squareroot
of (z-1)/(z+1).

I have used the quotient rule and cancelled out some stuff to get:
[z-1]/[2(z-1)^(.5) (z+1) - 2(z+1)^(.5) (z+1)]
The answer is supposed to be: 1/[(z-1)^(.5)(z+1)^(1.5)]
I can't seem to get to that point. Please help.

 

[skeeter]

\(\displaystyle \L \frac{d}{dz}(\frac{z-1}{z+1})^{\frac{1}{2}}\)

\(\displaystyle \L \frac{1}{2}(\frac{z-1}{z+1})^{-\frac{1}{2}}*\frac{(z+1) - (z-1)}{(z+1)^2}\)

\(\displaystyle \L \frac{1}{2}(\frac{z+1}{z-1})^{\frac{1}{2}}*\frac{2}{(z+1)^2}\)

\(\displaystyle \L \frac{\sqrt{z+1}}{\sqrt{z-1} (z+1)^2}\)

\(\displaystyle \L \frac{1}{\sqrt{(z-1)(z+1)^3}\)

 

[galactus]

Using the quotient rule and some algebra:

\(\displaystyle \L\\\frac{\sqrt{z+1}\frac{1}{2\sqrt{z-1}}-\sqrt{z-1}\frac{1}{2\sqrt{z+1}}}{z+1}\)

Do some algebraic somersaults and you will arrive at your answer. It's equivalent.

 

[sofiaahmed]

Thanks skeeter! Thanks galactus!