Using the definition of the derivative...

[johnjones]

I'm kind of stuck with this question:
"Use the limit definition of derivative to find f'(a) for f(x) = squareroot(1+x^2)."

I used the difference of squares method and ended up with a numerator of (a+h)^2 - a^2. The denominator is h(squareroot[1+[a+h]^2]+squareroot[1+a^2]). I took out an h in the numerator and in the denominator. Now, I'm kind of stuck with:

2a + h
----------
(squareroot[1+[a+h]^2])+squareroot[1+a^2])

Maybe there was another definition of the derivative I should have used... like the one as h -> 0. I'm not sure if I've used the right one, or the easier one. Can someone please help? thanks.

 

[Daniel_Feldman]

Let f(x) be a function and f'(x) denote its derivative.

The limit definition is:

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

In your case,

f(x) = squareroot(1+x^2)

f'(x)=lim (root(1+(x+h)^2)-root(1+x^2))/h
h->0


See if you can work from there.

 

[johnjones]

Let f(x) be a function and f'(x) denote its derivative.

The limit definition is:

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

In your case,

f(x) = squareroot(1+x^2)

f'(x)=lim (root(1+(x+h)^2)-root(1+x^2))/h
h->0


See if you can work from there.

I got (1+x^2+2h+h^2)^2 - (1+x^2)^2 {Used difference of squares}
----------------------------------------
h(sroot(1+(x+h)^2)+sroot(1+x^2).

I'm not sure what I can do next... I can't really cancel things at this point. Should I expand, divide?

 

[Daniel_Feldman]

Don't do that. The easiest thing to do in this case would be to multiply by the conjugate of the numerator.



(root(1+(x+h)^2)-root(1+x^2))/h multiplied by



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


That should eliminate the radicals in the numerator. Do not multiply out the denominator-leave it in factored form. Cancel out h's, and then take the limit as h->0 by substituting 0 in for h. Then simplify your result.