Question

Find the derivative of f(x) using the definition of a derivative.

Answer 1

using limit formula ? or by standard derivatives ?

Answer 2

Limit formula.

Answer 3

a weeks worth of algebra, and a rather sadistic question

Answer 4

No kidding!!!

Answer 5

\[f'(x) = \frac{f(x+\Delta{x})-f(x)}{\Delta{x}}\]

Answer 6

well not so bad if you start with
\[\frac{1}{\sqrt{x}}+x^2\]

Answer 7

I know that lol. But I cannot get rid of the h's. I got lost on simplyfing this monster,

Answer 8

and since the limit of the sum is the sum of the limits, you can do this in pieces

Answer 9

I have to do it as a single definition.

Answer 10

so.....

Answer 11

hahaha...

Answer 12

for example, the second one is standard and rather simple
\[\lim_{h\to 0}\frac{(x+h)^2-x^2}{h}=\lim_{x\to 0}\frac{x^2+2xh+h^2-x^2}{h}\]\[-\lim-{h\to 0}\frac{2xh+h^2}{h}=\lim_{x\to 0}2x+h=2x\] easy one

Answer 13

I get that but I have to use a single definition. No combining functions allowed.

Answer 14

I mean no seperating allowed.

Answer 15

\[\lim_{h\to 0}\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}\] might be harder, but not much

Answer 16

why there is a rule for not separating ??

Answer 17

Professor said so.

Answer 18

no matter, write it as one thing, then group them as first one as second one
the limit of the sum is the sum of the limits, it is the commutative law after all

Answer 19

so you can write it as one long piece if you like, but you will still combine like terms and get the same thing
it makes no difference, just looks messier

Answer 20

Hmm... good point.

Answer 21

Thanks! Makes this monster a lot easier to derive.