Derivative problem.

Chevy

New member
Joined
Jul 20, 2007
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I just started calc and I am stuck on a homework problem. Here are the instructions (I'm having trouble on the first part of it on this problem):
calcproblem1lw9.jpg


The actual problem:
calcproblem2bp7.jpg


I am using the Difference Quotient to figure it out.

And here is the work I've done:
calcproblem3vs7.jpg


The answer should get to be 1+(1/x^2) at some point (after making h = 0 at some point)

I realize I could simplify the denominator a little further or change the 1 to a fraction with the same denominator but I don't see it doing anything for me, so I am stuck and need help. :(
 
\(\displaystyle \L\\\lim_{h\to\0}\frac{(x+h-\frac{1}{x+h})-(x-\frac{1}{x})}{h}\)

This factors to:

=\(\displaystyle \L\\\lim_{h\to\0}\frac{x^{2}+xh+1}{x(x+h)}\)

I know, it's mostly the algebra that causes the trouble.

Now, as h approaches 0, what do you get?. I bet it's the derivative of
x-(1/x)

Once you have the derivative, then you have the slope. Sub in x=3 and get the slope at x=3. Then you can use y=mx+b to find the line equation.

x=3, y=8/3, the slope you'll have. Then solve for b and you're done.
 
Thank you very much, I guess I did the algebra completely wrong. :p I figured out how to factor the way you did. Once again, thanks for the help.
 
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