Home » Blog » Determine the value of g(x) = xf(x^2) and its derivatives given values of f(x)

Determine the value of g(x) = xf(x^2) and its derivatives given values of f(x)

by RoRi

Given a function g(x) = xf(x^2) and the following table of values for f and its derivatives:

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

Compute the value of g(x), g'(x) and g''(x) for x = 0,1,2.

First, by the chain rule we have

    \begin{align*} g'(x) &= f(x^2) + 2x^2 f'(x^2) \\ g''(x) &= 6xf'(x^2) + 4x^3 f''(x^2). \end{align*}

Using these formulas we compute the table,

    \[ \begin{tabular}{|c | c | c | c |} \hline $x$ & $g(x)$ & $g'(x)$ & $g''(x)$ \\ \hline 0 & 0 & 0 & 0 \\ 1 & 1 & 3 & 10 \\ 2 & 12 & 30 & 36 \\ \hline \end{tabular} \]

One comment

  1. José Ignacio Lobato says:

    dude, i think Apostol says g(x)=f(x^2). by the way its so helpful this solution book you’ve made, appreaciate that!!!!

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):