#### Answer

\[\frac{{{d^3}}}{{d{x^3}}} = 29.568\]

#### Work Step by Step

\[\begin{gathered}
{\left. {\frac{{{d^3}}}{{d{x^3}}}\,\left( {{x^{4.2}}} \right)} \right|_{x = 1}} \hfill \\
\hfill \\
differentiate\,\,use\,\,the\,\,power\,\,rule \hfill \\
\hfill \\
\frac{d}{{dx}} = 4.2{x^{4.2 - 1}} = 4.2{x^{3.2}} \hfill \\
\hfill \\
differentiate\,\,use\,\,the\,\,power\,\,rule{\text{ again}} \hfill \\
\hfill \\
\frac{{{d^2}}}{{d{x^2}}} = \,\left( {4.2} \right)\,\left( {3.2} \right){x^{3.2 - 1}} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{{d^2}}}{{d{x^2}}} = 13.44{x^{2.2}} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{{d^3}}}{{d{x^3}}} = \,\left( {2.2} \right)\,\left( {13.44} \right){x^{2.2 - 1}} \hfill \\
\hfill \\
Simplify \hfill \\
\hfill \\
\frac{{{d^3}}}{{d{x^3}}} = 29.568{x^{1.1}} \hfill \\
\hfill \\
Evaluate\,\,\,{\text{ at}}\;\;{\text{ }}x = 1 \hfill \\
\hfill \\
\frac{{{d^3}}}{{d{x^3}}} = 29.568{\left( 1 \right)^{1.1}} \hfill \\
\hfill \\
\frac{{{d^3}}}{{d{x^3}}} = 29.568 \hfill \\
\end{gathered} \]