Question

If a function is given by $f\left(x\right)=\frac{x+3}{x^2-1}$, what is the instantaneous rate of change of the function at $x=3$?

• A. $\frac{7}{16}$
• B. $-\frac{7}{16}$
• C. $\frac{11}{16}$
• D. $-\frac{11}{16}$
• E. $\frac{1}{6}$

Answer 1

Answer: (B)

$-\frac{7}{16}$

Given, $f\left(x\right)=\frac{x+3}{x^2-1}$
The instantaneous rate of function is $f^{\prime }\left(x\right)=\frac{(x^2-1)\left(1\right)-(x+3)\left(2x\right)}{{(x^2-1)}^2}=\frac{x^2-1-2x^2-6x}{{(x^2-1)}^2}=\frac{-x^2-1-6x}{{(x^2-1)}^2}$
Now put $x=3$, we get $f^{\prime }\left(3\right)=\frac{-9-1-18}{64}=-\frac{28}{64}=-\frac{7}{16}$