Question

1. use the table to calculate $c(1)$ for $c(x)=f(g(x))

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$$\begin{array}{|c|c|c|c|c|} \\hline x&f(x)&g(x)&f(x)&g(x)\\ \\hline 1&3&4&1&2\\ \\hline 2&5&7&2&6\\ \\hline 3&12&9&8& - 1\\ \\hline 4&21&8&10& - 3\\ \\hline \\end{array}$$

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:

$20$