Definition of Derivative

antinerd

1. The problem statement, all variables and given/known data
Find the derivative using the Definition of the Derivative:

f(x) = 1 / x^2


2. Relevant equations

The Definition:

f`(a) = lim h->0 [f(a+h) - f(a)] / h

3. The attempt at a solution

This is what I did:

f`(a) = lim h->0 [tex](1/(x+h)^{2}) - 1/x^{2}) / (h)[/tex]

f`(a) = lim h->0 [tex][((x^{2}) - 1 (x^{2} + 2xh + h^{2})) / x^{2}(x^{2} + 2xh + h^{2})] / h[/tex]

f`(a) = lim h->0 [tex](2x + h) / (x^{4} + 2x^{3}h + x^{2}h^{2})[/tex]

and finally

f`(a) = lim h->0 [tex]2 / x^{3}[/tex]

So I got that as the derivative, and if I did it correctly it should be right. Did I use the definition properly?

antinerd

lol nevermind im a retard i figured it out... sigh so much typing for nothing

Feldoh

Should be

[tex]f'(a) = \lim_{h \to 0} \frac{-(2x + h)}{(x^{4} + 2x^{3}h + x^{2}h^{2})}[/tex]

Also when you get to the final answer [itex]\frac{-2}{x^3}[/itex] you already took the limit so the answer is just:

[tex]f'(a) = \frac{-2}{x^3}[/tex]

Because:
[tex]f'(a) = \lim_{h \to 0} \frac{-(2x + 0)}{(x^{4} + 2x^{3}*0 + x^{2}*0^{2})}[/tex]
[tex]f'(a) = \frac{-2x}{x^{4}}[/tex]