Question

f(x) = 2011/x^2. Use the DEFINITION of the derivative to compute f'(2). I keep getting 0 on top. What am I doing wrong?

Answer 1

The difference quotient
(f(x) - f(a))/(x-a) = 2011 [ 1/x^2 - 1/4 ] / (x - 2)

Answer 2

Now, that equals\[2011 \frac{1}{4x^2} \frac{ 4 - x^2}{x - 2} = 2011 \frac{1}{4x^2} \frac{ (2-x)(2+x)}{x-2}\]

Answer 3

Now cancel 2 - x and we're left with ...

-(2011/4x^2) (x + 2)

Answer 4

And taking the limit as x --> 2 of this now

-2011/16 = -2011/2 . (1/2^3)

...which is what we would expect from apply the standard rule to the function in the first place.

Answer 5

Okay, I get the first step now, but I don't understand how you got from the first one to the second

Answer 6

I took 4x^2 as a common denominator for the terms in the numerator,
1/x^2 - 1/4 = (1/4x^2) ( 4 - x^2 )

Answer 7

but wouldn't you go: 1/x^2 (4/4) and 1/4(x^2/x^2) to make a common denominator. making this 4-x^2/4x^2

Answer 8

Look at it carefully -- you have written down the same thing.

Answer 9

aaah. okay. thank you :) I hate when profs ask for the long route :/

Answer 10

It's to make you feel comfortable that these slightly magical formulas for derivative are really working and to teach you to keep in your head what the derivative really is.

It's not that we would ever in practice calculate it this way, but it's good to know and feel comfortable and indeed certain that when we do use a formula, it gives the same result as the basic definition.

Answer 11

So now, do yourself a favor. Take a blank piece of paper and work the entire problem again from the beginning.

Then take another piece of blank piece of paper and work the entire problem without looking at the first piece or our discussion here.

When you can do that, then you really understand this problem.

Answer 12

Haha I did that before you even said it :) Thank you again.

Answer 13

Very good :-)