In each part, consider the function \(\displaystyle f\) that is defined and calculate \(\displaystyle \left(Df\right)_p (h)\), \(\displaystyle \left(Df\right)^2_p (h,k)\), and \(\displaystyle \left(Df\right)^3_p (h,k,l)\) for:
\(\displaystyle p\, =\, \left(\begin{array}{c}x\\y\end{array}\right)\, \mbox{ and }\, p\, =\, \left(\begin{array}{c}1\\0\end{array}\right)\)
where
\(\displaystyle h\, =\, \left(\begin{array}{c}h_1\\h_1\end{array}\right),\, k\, =\, \left(\begin{array}{c}k_1\\k_1\end{array}\right),\, \mbox{ and }\, l\, =\, \left(\begin{array}{c}l_1\\l_1\end{array}\right)\)
are points in \(\displaystyle \mathbb{R}^2\).
a) \(\displaystyle f\, :\, \mathbb{R}^2\, \rightarrow\, \mathbb{R}\) is defined by \(\displaystyle f\, \left(\begin{array}{c}x\\y\end{array}\right)\, =\, x^3\, +\, 2xy\, -\, e^y\)
b) \(\displaystyle f\, :\, \mathbb{R}^2\, \rightarrow\, \mathbb{R}^2\) is defined by \(\displaystyle f\, \left(\begin{array}{c}x\\y\end{array}\right)\, =\, \left(\begin{array}{c}x^3\, +\, 2xy\, -\, e^y\\xy\, =\, 1\end{array}\right)\)