Question

let $r(x)=f(g(h(x)))$, where $h(1)=2$, $g(2)=5$, $h(1)=3$, $g(2)=3$, and $f(5)=5$. find $r(1)$. $r(1)=square$

Explanation:

Step1: Apply chain - rule

The chain - rule for a composite function $r(x)=f(g(h(x)))$ is $r^{\prime}(x)=f^{\prime}(g(h(x)))\cdot g^{\prime}(h(x))\cdot h^{\prime}(x)$.

Step2: Evaluate at $x = 1$

Substitute $x = 1$ into $r^{\prime}(x)$. We get $r^{\prime}(1)=f^{\prime}(g(h(1)))\cdot g^{\prime}(h(1))\cdot h^{\prime}(1)$.

Step3: Use given values

Since $h(1)=2$, then $g(h(1))=g(2)=5$ and $g^{\prime}(h(1))=g^{\prime}(2)=3$. Also, $h^{\prime}(1)=3$ and $f^{\prime}(5)=5$.
So $r^{\prime}(1)=f^{\prime}(5)\cdot g^{\prime}(2)\cdot h^{\prime}(1)$.

Step4: Calculate the result

Substitute the values $f^{\prime}(5)=5$, $g^{\prime}(2)=3$, and $h^{\prime}(1)=3$ into the expression. Then $r^{\prime}(1)=5\times3\times3 = 45$.

Answer:

$45$