Quick Calculus Help

[JohnCU]

I'm trying to take the derivative of f(x) = x + sqrt(x) the long way...

The limit as h -> 0 of (f(x+h) - f(x))/h ...

= as h -> 0, (h + sqrt(x+h) - sqrt(x)) / h

I'm stuck here, I don't know how to simplify this to cancel out the h. If the h wasn't on the top, I could multiply the top and bottom by (sqrt(x+h) + sqrt(x)) to cancel out the roots...

 

[FeathersMcGraw]

Originally posted by: JohnCU

I'm stuck here, I don't know how to simplify this to cancel out the h. If the h wasn't on the top, I could multiply the top and bottom by (sqrt(x+h) + sqrt(x)) to cancel out the roots...

Go ahead and do that, then simplify the resulting expression.

 

[notfred]

1+ (1/2)x^(-1/2)

 

[Legendary]

You can do what you want to do (multiply by roots) even if there's an h on top. Just make sure you multiply properly.

 

[JohnCU]

Well I still can't get it to work, the answer the ti-89 gives me is like (1/sqrt(2x)) + 1 but no matter what I multiply the top by to get rid of the roots, I still can't get the answer.

 

[FeathersMcGraw]

Originally posted by: JohnCU
Well I still can't get it to work, the answer the ti-89 gives me is like (1/sqrt(2x)) + 1 but no matter what I multiply the top by to get rid of the roots, I still can't get the answer.

You can't get rid of all the square roots, since they're part of the answer. However, if you multiply both numerator and denominator by [sqrt(x + h) + sqrt(h)] as you originally mentioned, you should be able to simplify and cancel some of the x and h factors until you can safely take the limit as h approaches zero.

 

[JohnCU]

(h+sqrt(x+h) - sqrt(x)) * (sqrt(x+h) + sqrt(x))

=

lim as h->0 of h*sqrt(x+h)+h*sqrt(x)+h divided by h*(sqrt(x+h) + sqrt(x))

and that doesn't come out to 1/(2*sqrt(x)) + 1...

 

[FeathersMcGraw]

Originally posted by: JohnCU
(h+sqrt(x+h) - sqrt(x)) * (sqrt(x+h) + sqrt(x))

=

lim as h->0 of h*sqrt(x+h)+h*sqrt(x)+h divided by h*(sqrt(x+h) + sqrt(x))

and that doesn't come out to 1/(2*sqrt(x)) + 1...

It really does.

It might be more visible if you multiply numerator and denominator by [sqrt(x + h) + sqrt(x)] after you've simplified [h + sqrt(x + h) - sqrt(x)]/h to 1 + [sqrt(x + h) - sqrt(x)]/h.

 

[JohnCU]

Anybody else wanna give it a shot?

I've worked it out by all the suggestions and using the ti-89 and the closest I get is [2*sqrt(x) + 1] / [2*sqrt(x)]

 

[FeathersMcGraw]

Originally posted by: JohnCU
Anybody else wanna give it a shot?

I've worked it out by all the suggestions and using the ti-89 and the closest I get is [2*sqrt(x) + 1] / [2*sqrt(x)]

Which is pretty much the answer you got before if you divide numerator and denominator by 2 * sqrt(x).