use the defiition of derivitive to compute f'(3)
f(x)=1+(1/(x-1))

Question

use the defiition of derivitive to compute f'(3)
f(x)=1+(1/(x-1))

anonymous · Answer

pleasee hurry!!

anonymous · Answer

$$f(x)=1+\frac{1}{x-1}$$
By the limit definition of the derivative,
$$\begin{align*}f'(x)&=\lim_{h\to0}\frac{\left(1+\dfrac{1}{x+h-1}\right)-\left(1+\dfrac{1}{x-1}\right)}{h}\\
&=\lim_{h\to0}\frac{\dfrac{1}{x+h-1}-\dfrac{1}{x-1}}{h}\\
&=\lim_{h\to0}\frac{\dfrac{x-1}{(x-1)(x+h-1)}-\dfrac{x+h-1}{(x-1)(x+h-1)}}{h}\\
&=\lim_{h\to0}\frac{x-1-(x+h-1)}{h(x-1)(x+h-1)}\\
&=\lim_{h\to0}\frac{-h}{h(x-1)(x+h-1)}\\
&=\lim_{h\to0}\frac{-1}{(x-1)(x+h-1)}\\
&=-\frac{1}{(x-1)^2}
\end{align*}$$
Plug in $x=3$ to find $f'(3)$.