# derivative: can't get rid of 1/h

• January 19th 2013, 10:23 AM
infraRed
derivative: can't get rid of 1/h
Hi. I'm trying to review stuff I learn in Calc I, and I'm up to derivatives. However, I'm stumped on this prob from my old book:

find derivative of 1/sqrt(x+2)

After putting it in the form lim(h->0) [(1/h)(f(x+h)-f(x))], substituting, and doing some algebra I end up with

lim(h->0)[(1/h)((sqrt(x+2)-sqrt(x+h+2))/(sqrt(x+h+2)*sqrt(x+2)))]

If I multiple by conjugate I get 2x+h+4 on the top, and something really compilicated on the bottom, and I'm still not clear on how to get rid of the 1/h. Please advise me on the correct approach.
• January 19th 2013, 11:30 AM
HallsofIvy
Re: derivative: can't get rid of 1/h
Quote:

Originally Posted by infraRed
Hi. I'm trying to review stuff I learn in Calc I, and I'm up to derivatives. However, I'm stumped on this prob from my old book:

find derivative of 1/sqrt(x+2)

After putting it in the form lim(h->0) [(1/h)(f(x+h)-f(x))], substituting, and doing some algebra I end up with

lim(h->0)[(1/h)((sqrt(x+2)-sqrt(x+h+2))/(sqrt(x+h+2)*sqrt(x+2)))]

If I multiple by conjugate I get 2x+h+4 on the top, and something really compilicated on the bottom, and I'm still not clear on how to get rid of the 1/h. Please advise me on the correct approach.

You don't get that if you multiply correctly! [tex](\sqrt{x+2}- \sqrt{x+ h+ 2})(\sqrt{x+ 2}+ \sqrt{x+ h+ 2})= (x+ 2)- (x+ h+ 2)= -h. You added instead of subtracting.

The denominator will be $(\sqrt{x+ 2}+ \sqrt{x+ h+ 2})\sqrt{x+ 2}{\sqrt{x+h+ 2}$. Yes, that's complicated but now you can let h go to 0.
• January 19th 2013, 11:48 AM
infraRed
Re: derivative: can't get rid of 1/h
Oops. Duh! I can't count the number of times a mishandled sign has cost me half-an-hour in homework time. Thanks for the help!
• January 19th 2013, 11:59 AM
hollywood
Re: derivative: can't get rid of 1/h
Quote:

Originally Posted by infraRed
Oops. Duh! I can't count the number of times a mishandled sign has cost me half-an-hour in homework time. Thanks for the help!

You're not alone.

- Hollywood