Can someone help me finish this problem?

[Joey29]

Hey, I have my exam this week, and a review problem asks me to differentiate this without using rules. :evil:

So...

f'(x) = lim f(x+h) -f(x)
h>0 h

f(x) = x ^1/2

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

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

I'm stuck there, can someone finish this problem...I know I have to factor out an "h" to cancel the h on the top and bottom but how...maybe thats not the problem. I'm not sure. Please respond asap. I appreciate any help!!!
Joe

 

[galactus]

You're almost there.

\(\displaystyle \frac{(sqrt{x+h}-sqrt{x})(sqrt{x+h}+sqrt{x})}{h(sqrt{x+h}+sqrt{h})}\)

=\(\displaystyle \frac{(x+h)-x}{h(sqrt{x+h}+sqrt{x})}\)

=\(\displaystyle \frac{h}{h(sqrt{x+h}+sqrt{x})}\)

=\(\displaystyle \frac{1}{sqrt{x+h}+sqrt{x}}\)

Since \(\displaystyle {h\to\0}\):

=\(\displaystyle \frac{1}{sqrt{x}+sqrt{x}}=\frac{1}{\2sqrt{x}}\)

 

[Joey29]

Thankyou Galactu!!!!

Thanks!! It looked so ugly I thought it wasn't possible!