Question

for each problem, find the instantaneous rate of change of the function at the given value. 7) $f(x)=-\frac{1}{x + 1}$; 0

Explanation:

Step1: Find the derivative

The function is $f(x)=-\frac{1}{x + 1}=-(x + 1)^{-1}$. Using the power - rule for differentiation $\frac{d}{dx}(u^n)=nu^{n - 1}\frac{du}{dx}$, where $u=x + 1$ and $n=-1$. So $f^\prime(x)=(-1)\times(-1)\times(x + 1)^{-2}=\frac{1}{(x + 1)^2}$.

Step2: Evaluate the derivative at the given point

We need to find the instantaneous rate of change at $x = 0$. Substitute $x = 0$ into $f^\prime(x)$. So $f^\prime(0)=\frac{1}{(0 + 1)^2}=1$.

Answer:

1