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Compute values of a composite function given values of the components

Given functions f(x) and g(x) with the following values

    \[ \begin{tabular}{|c | c | c | c | c |}  \hline  $x$ & $f(x)$ & $f'(x)$ & $g(x)$ & $g'(x)$ \\  \hline   0 & 1 & 5 & 2 & -5 \\  1 & 3 & -2 & 0 & 1 \\  2 & 0 & 2 & 3 & 1 \\  3 & 2  & 4 & 1 & -6 \\  \hline \end{tabular} \]

Let h(x) = f(g(x)) and k(x) = g(f(x)) and construct a table as above for the functions h and k.


First, by the chain rule we have

    \begin{align*}  h(x) &= f(g(x)) &\implies \quad h'(x) &= f'(g(x)) g'(x) \\  k(x) &= g(f(x)) &\implies \quad k'(x) &= g'(f(x)) f'(x). \end{align*}

Using these formulas we compute the table,

    \[ \begin{tabular}{|c | c | c | c | c |}  \hline  $x$ & $h(x)$ & $h'(x)$ & $k(x)$ & $k'(x)$ \\  \hline   0 & 0 & -10 & 0 & 5 \\  1 & 1 & 5 & 1 & 12 \\  2 & 2 & 4 & 2 & -10 \\  3 & 3  & 12 & 3 & 4 \\  \hline \end{tabular} \]

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