Differentiation - One question per page

15.3 Section 13.4 Progress Check Question 3

Let \(f(x,y) = (xy,x, y^3)\) and \(g(u,v)=(u^2+1,u-v^2)\). Calculate \({\bf D} (f\circ g)(1,1)\) using the chain rule.

\(\ \ \displaystyle {\bf D} (f\circ g)(1,1) = \left[ \begin{array}{ccc} 2&2&0\\ 2&0&4 \end{array} \right] \)

\(\ \ \displaystyle {\bf D} (f\circ g)(1,1) = \left[ \begin{array}{ccc} 3&2&1\\ -2&0&-2 \end{array} \right] \)

\(\ \ \displaystyle {\bf D} (f\circ g)(1,1) = \left[ \begin{array}{c@{\quad}c} 2&0\\ 4&0\\ 0&0 \end{array} \right] \)

\(\ \ \displaystyle {\bf D} (f\circ g)(1,1) = \left[ \begin{array}{c@{\quad}c} 2&-4\\ 2&0\\ 0&0\end{array} \right] \)

\(\ \ \displaystyle {\bf D} (f\circ g)(1,1) = \left[ \begin{array}{c@{\quad}c} 3&-2\\ 2&0\\ 1&-2 \end{array} \right] \)

Correct.

Try again. Remember to evaluate \({\bf D}f\) at \(g(1,1)\).

Incorrect.