Question

estimate the instantaneous rate of change of $y(x)=\frac{5}{x + 1}$ at the point $x = 1$. your answer should be accurate to at least 3 decimal places.

Explanation:

Step1: Recall the quotient - rule for differentiation

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 5$, $u^\prime=0$, $v=x + 1$, and $v^\prime = 1$.

Step2: Apply the quotient - rule

$y^\prime(x)=\frac{0\times(x + 1)-5\times1}{(x + 1)^{2}}=-\frac{5}{(x + 1)^{2}}$.

Step3: Evaluate the derivative at $x = 1$

Substitute $x = 1$ into $y^\prime(x)$. $y^\prime(1)=-\frac{5}{(1 + 1)^{2}}=-\frac{5}{4}=-1.250$.

Answer:

$-1.250$