Question

(1 point) let ( r ( x ) = f ( g ( h ( x ) ) ) ), where ( h ( 1 ) = 2 ), ( g ( 2 ) = 3 ), ( h ^ { prime } ( 1 ) = 4 ), ( g ^ { prime } ( 2 ) = 5 ), and ( f ^ { prime } ( 3 ) = 6 ). find ( r ^ { prime } ( 1 ) ). ( r ^ { prime } ( 1 ) = )

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(r(x)=f(g(h(x)))\), then \(r^{\prime}(x)=f^{\prime}(g(h(x)))\cdot g^{\prime}(h(x))\cdot h^{\prime}(x)\).

Step2: Substitute \(x = 1\)

When \(x = 1\), we have:
\(r^{\prime}(1)=f^{\prime}(g(h(1)))\cdot g^{\prime}(h(1))\cdot h^{\prime}(1)\)
Since \(h(1) = 2\), then \(r^{\prime}(1)=f^{\prime}(g(2))\cdot g^{\prime}(2)\cdot h^{\prime}(1)\)

Step3: Substitute known values

Given \(g(2)=3\), \(g^{\prime}(2) = 5\), \(h^{\prime}(1)=4\), and \(f^{\prime}(3)=6\)
\(r^{\prime}(1)=f^{\prime}(3)\cdot5\cdot4\)
\(r^{\prime}(1)=6\times5\times4\)
\(r^{\prime}(1)=120\)

Answer:

\(120\)