Answer:
[tex]\begin{gathered} m=M(\frac{19}{10})=\frac{57\sqrt[10]{10}\cdot19^{\frac{9}{10}}}{50} \\ m=20.313 \end{gathered}[/tex]Step-by-step explanation:
The slope of the tangent line is given at the first derivate of the function, evaluated at x=x0. The value of the function at the given point:
[tex]\begin{gathered} p(x)=6(1.9)^{1.9} \\ p(\frac{19}{10})=\frac{57\sqrt[10]{10}\cdot19^{\frac{9}{10}}}{50} \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} p(x)=6x^{1.9} \\ p^{\prime}(x)=\frac{57x^{\frac{9}{10}}}{5} \end{gathered}[/tex]Then, for the following function the slope of the tangent line is:
[tex]\begin{gathered} M(x)=p^{\prime}(x)=\frac{57x^{\frac{9}{10}}}{5} \\ \text{Then,} \\ m=M(\frac{19}{10})=\frac{57\sqrt[10]{10}\cdot19^{\frac{9}{10}}}{50} \end{gathered}[/tex]