# limit definition of derivative

• Mar 3rd 2011, 12:48 PM
elieh
limit definition of derivative
3)Use the limit definition of derivative and find, $\displaystyle y'$
for $\displaystyle \frac{1}{x+1}$

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.
• Mar 3rd 2011, 12:54 PM
Plato
Find this limit.
$\displaystyle \displaystyle\lim _{h \to 0} \frac{{\frac{1} {{x + 1 + h}} - \frac{1} {{x + 1}}}} {h}$
• Mar 3rd 2011, 01:02 PM
elieh
I end up with:

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

Originally Posted by elieh
I end up with:$\displaystyle \frac{h^2}{(x+1)^2 +h(x+1)}$

$\displaystyle \dfrac{\frac{1}{x+h+1}-\frac{1}{x+1}}{h}=\dfrac{\frac{(x+1)-(x+h+1)}{(x+h+1)(x+1)}}{h}$
• Mar 3rd 2011, 01:22 PM
elieh
$\displaystyle \frac{-h^2}{(x+1)^2 + h(x+1)}$

If h is on top and it tends to 0 then the limit is 0...
• Mar 3rd 2011, 01:41 PM
Plato
You said that you already know the answer: $\displaystyle \dfrac{-1}{(x+1)^2}$