7. use the table to calculate $c(1)$ for $c(x)=f(g(x))\n\\begin{array}{|c|c|c|c|c|}\n\\hline…
7. use the table to calculate $c(1)$ for $c(x)=f(g(x))\n\\begin{array}{|c|c|c|c|c|}\n\\hline x&f(x)&g(x)&f(x)&g(x)\\\n\\hline 1&3&4&1&2\\\n\\hline 2&5&7&2&6\\\n\\hline 3&12&9&8& - 1\\\n\\hline 4&21&8&10& - 3\\\n\\hline\n\\end{array}
Answer
Explanation:
Step1: Apply chain - rule
The chain - rule states that if $c(x)=f(g(x))$, then $c^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$.
Step2: Evaluate at $x = 1$
We want to find $c^{\prime}(1)$. First, when $x = 1$, $g(1)=4$. Then $f^{\prime}(g(1))=f^{\prime}(4)$ and $g^{\prime}(1)=2$. From the table, $f^{\prime}(4) = 10$. So $c^{\prime}(1)=f^{\prime}(g(1))\cdot g^{\prime}(1)$.
Step3: Calculate the result
Substitute the values: $c^{\prime}(1)=10\times2=20$.
Answer:
$20$