limit definition of derivative

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March 3rd 2011, 12:48 PM
elieh

limit definition of derivative

3)Use the limit definition of derivative and find, $y'$
for $\frac{1}{x+1}$

I know have the answer because we are taught the short cut method in school where we multiply the coefficient by the power and decrease by one. But I want to see how this is done the longer method.
March 3rd 2011, 12:54 PM
Plato

Find this limit.
$\displaystyle\lim _{h \to 0} \frac{{\frac{1}<br /> {{x + 1 + h}} - \frac{1}<br /> {{x + 1}}}}<br /> {h}$
March 3rd 2011, 01:02 PM
elieh

I end up with:

$\frac{h^2}{(x+1)^2 +h(x+1)}$
March 3rd 2011, 01:14 PM
Plato

Quote:

Originally Posted by elieh

I end up with: $\frac{h^2}{(x+1)^2 +h(x+1)}$

You made some algebra mistakes.
$\dfrac{\frac{1}{x+h+1}-\frac{1}{x+1}}{h}=\dfrac{\frac{(x+1)-(x+h+1)}{(x+h+1)(x+1)}}{h}$
March 3rd 2011, 01:22 PM
elieh

$\frac{-h^2}{(x+1)^2 + h(x+1)}$

If h is on top and it tends to 0 then the limit is 0...
March 3rd 2011, 01:41 PM
Plato

The answer is not zero.
Look at post #4. Can you do the algebra?
You said that you already know the answer: $\dfrac{-1}{(x+1)^2}$
So work towards that answer.
March 3rd 2011, 01:53 PM
elieh

Got it thanks!

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