To find [tex]\(h'(-1)\)[/tex] where [tex]\(h(x) = \frac{3}{[g(x)]^4}\)[/tex], let's follow these steps:
1. Express [tex]\(h(x)\)[/tex] explicitly:
[tex]\[
h(x) = 3 [g(x)]^{-4}
\][/tex]
2. Compute the derivative [tex]\(h'(x)\)[/tex] using the chain rule:
[tex]\[
h(x) = 3 [g(x)]^{-4}
\][/tex]
Let [tex]\(u = g(x)\)[/tex]. Then [tex]\(h(x) = 3 u^{-4}\)[/tex].
Compute the derivative [tex]\(\frac{d}{dx}[3 u^{-4}]\)[/tex]:
[tex]\[
\frac{d}{dx}[3 u^{-4}] = 3 \cdot (-4) u^{-5} \frac{du}{dx}
\][/tex]
[tex]\[
\frac{d}{dx}[3 u^{-4}] = -12 u^{-5} \frac{du}{dx}
\][/tex]
3. Substitute back [tex]\(u = g(x)\)[/tex] and the chain rule result:
[tex]\[
h'(x) = -12 [g(x)]^{-5} g'(x)
\][/tex]
4. Evaluate [tex]\(h'(-1)\)[/tex] using the given values from the table:
From the table:
[tex]\[
g(-1) = 6
\][/tex]
[tex]\[
g'(-1) = 1
\][/tex]
Substitute [tex]\(g(-1)\)[/tex] and [tex]\(g'(-1)\)[/tex] into the derivative formula:
[tex]\[
h'(-1) = -12 [g(-1)]^{-5} g'(-1)
\][/tex]
5. Calculate [tex]\(h'(-1)\)[/tex]:
[tex]\[
h'(-1) = -12 [6]^{-5} \cdot 1
\][/tex]
[tex]\[
h'(-1) = -12 \cdot 6^{-5} \cdot 1
\][/tex]
6. Simplify [tex]\(6^{-5}\)[/tex]:
[tex]\[
6^{-5} = \frac{1}{6^5} = \frac{1}{7776}
\][/tex]
[tex]\[
h'(-1) = -12 \cdot \frac{1}{7776}
\][/tex]
[tex]\[
h'(-1) = -\frac{12}{7776} = -\frac{1}{648} \approx -0.001543
\][/tex]
7. Round the answer to 3 decimal places:
[tex]\[
h'(-1) \approx -0.002
\][/tex]
So, the value of [tex]\(h'(-1)\)[/tex] is [tex]\(-0.002\)[/tex].
