Write an equation of the line tangent to the curve y=(1/x-1/x^2)^-1 at point (2,4). You must use the limit definition of the derivative...I have no idea where to start on this, just started derivatives in class

Question

whpalmer4 · Answer

The limit definition is:
$$f'(x) = \lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$$

You have $$y = f(x) =\frac{1}{1/x-1/x^2} = \frac{1}{x/x^2-1/x^2} = \frac{x^2}{x-1}$$Your mission, should you choose to accept it, is to find the limit of 
$$\lim_{h\rightarrow0}\frac{\frac{(x+h)^2}{x+h-1}-\frac{x^2}{x-1}}{h}$$ Your algebra skills will get a bit of a workout :-)

anonymous · Answer

I ended up with lim as h->0 being x^2-2x?

whpalmer4 · Answer

I don't think that's right...let me check

whpalmer4 · Answer

You've got the numerator correct, but there should be a denominator...

whpalmer4 · Answer

After you grind through it you should end up with
$$\lim_{h\rightarrow0}\frac{h (x-1)+(x-2) x}{(x-1) (x+h-1)}$$which you can evaluate directly.

anonymous · Answer

Alright well I definitely did get a bit of an algebraic workout but I think I got it now! Thank you!

whpalmer4 · Answer

What's your final result?

whpalmer4 · Answer

Yeah, these limits definitely are tedious exercises — you'll be glad to learn other ways to find them :-)