Question

Using the definition of a derivative....

Answer 1

....?

Answer 2

Supposed to use first principles to compute the derivatitve, but I am not sure how when f(x) is a piecewise function.

Answer 3

compute the derivative of the top, the derivative of the bottom, and see if they agree at 4

Answer 4

i agree with satellite

Answer 5

or else i guess you could compute
\[\lim_{x\to 4}\frac{\sqrt{x-4}-2}{x-4}\]

Answer 6

It would be easier to differentiate them individually I think. (For me at least)

Answer 7

thanks for the tip!

Answer 8

I'm not really suer how you got that limit, so I think i'll just try it the first way you suggested.

Answer 9

in any case if you just need the answer and don't have to write the work out by hand, then take the derivative separately, and then replace \(x\) by 4 and see if they agree

Answer 10

or you can work from the definition
\[\f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\]

Answer 11

I do have to show all work..

Answer 12

but you would have to do it twice with \(a=4\) first one would look like
\[\lim_{h\to 0}\frac{\sqrt{4+h}-0}{h}\] which does not exist

Answer 13

think about it this way
\[f(x)=\sqrt{4-x}\]
\[f'(x)=\frac{-1}{2\sqrt{4-x}}\] and this is not differentiable at \(x=4\) in any case

Answer 14

okay thank you.. another quick question... if a function f(x) exists at c, is it differentiable at x=c?

Answer 15

hell no

Answer 16

thanks :)

Answer 17

i mean it might be, but it might not be
even if it is continuous it need not be differentiable