Knowledge Check

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$, where $h \neq$
\[
f(x)=\frac{4}{x-1}
\]

Simplify your answer as much as possible.
\[
\frac{f(x+h)-f(x)}{h}=\square
\]

Answer

Expert–verified

Hide Steps

Answer

Simplify the difference quotient to get the final answer: $\boxed{-\frac{4}{(x - 1)(h + x - 1)}}$

Steps

Step 1 ：Define the function $f(x)=\frac{4}{x-1}$

Step 2 ：Substitute $x+h$ into the function for $x$, getting $f(x+h)=\frac{4}{x+h-1}$

Step 3 ：Subtract the original function $f(x)$ from $f(x+h)$, resulting in $f(x+h)-f(x)=\frac{4}{x+h-1}-\frac{4}{x-1}$

Step 4 ：Divide the result by $h$, getting the difference quotient $\frac{f(x+h)-f(x)}{h}=\frac{\frac{4}{x+h-1}-\frac{4}{x-1}}{h}$

Step 5 ：Simplify the difference quotient to get the final answer: $\boxed{-\frac{4}{(x - 1)(h + x - 1)}}$