Question2. Compute the derivative $D_{f}$ for each of the following functio (a) $f(t)=\begin{array}{c}t^{2} \\ t \cos (t)\end{array}$ (b) $f(x, y)=x^{2} y^{3} \cos (x y)$ (c) $f(x, y)=\left[\begin{array}{c}x+y \\ x y \\ x^{2} e^{y}\end{array}\right]$

Studdy Solution
Calculate each partial derivative:
1. $\frac{\partial}{\partial x}(x+y) = 1$, $\frac{\partial}{\partial y}(x+y) = 1$
2. $\frac{\partial}{\partial x}(xy) = y$, $\frac{\partial}{\partial y}(xy) = x$
3. $\frac{\partial}{\partial x}(x^2 e^y) = 2xe^y$, $\frac{\partial}{\partial y}(x^2 e^y) = x^2 e^y$
Thus, the Jacobian matrix is:
$$D_f(x, y) = \begin{bmatrix} 1 & 1 \\ y & x \\ 2xe^y & x^2 e^y \end{bmatrix}$$
The derivatives for each part are:
(a) $D_f(t) = \begin{array}{c} 2t \\ \cos(t) - t \sin(t) \end{array}$
(b) $\frac{\partial f}{\partial x} = 2xy^3 \cos(xy) - x^3y^4 \sin(xy)$, $\frac{\partial f}{\partial y} = 3x^2y^2 \cos(xy) - x^3y^4 \sin(xy)$
(c) $D_f(x, y) = \begin{bmatrix} 1 & 1 \\ y & x \\ 2xe^y & x^2 e^y \end{bmatrix}$