Question

if (f(x)=\frac{1}{(x + 1)^5}), find (f(x)) and then compute (f(-2)). use exact values.

(f(x)=)

(f(-2)=)

question help: video

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{1}{(x + 1)^5}=(x + 1)^{-5}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = u^n$, then $y^\prime=nu^{n - 1}u^\prime$. Here $u=x + 1$, $n=-5$, and $u^\prime = 1$. So $f^\prime(x)=-5(x + 1)^{-5-1}\times1=-5(x + 1)^{-6}=-\frac{5}{(x + 1)^6}$.

Step3: Evaluate $f^\prime(-2)$

Substitute $x=-2$ into $f^\prime(x)$. We get $f^\prime(-2)=-\frac{5}{(-2 + 1)^6}=-\frac{5}{(-1)^6}=-5$.

Answer:

$f^\prime(x)=-\frac{5}{(x + 1)^6}$
$f^\prime(-2)=-5$