a) $$ \frac { \partial }{ \partial x } =(\begin{matrix} cos(x)−xsin(x) \\ sin(x)+xcos(x) \end{matrix})=Df(x) $$
b) $$ \frac { \partial }{ \partial { x }_{ 1 }} =(−{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 })−cos({ x }_{ 1 }+{ x }_{ 2 }))),\quad \frac { \partial }{ \partial { x }_{ 2 } } =(−{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 }))\quad \Longrightarrow \\Df(x)=(\begin{matrix} −{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 })−cos({ x }_{ 1 }+{ x }_{ 2 })) \\ −{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 }) \end{matrix}) $$
c) $$ \frac { \partial }{ \partial { x }_{ 1 } } ={ { x }_{ 2 }{ x }_{ 3 }e }^{ { x }_{ 1 }{ x }_{ 2 } }\quad \frac { \partial }{ \partial { x }_{ 2 } } ={ { x }_{ 1 }{ x }_{ 3 }e }^{ { x }_{ 1 }{ x }_{ 2 } }\quad \frac { \partial }{ \partial { x }_{ 3 } } ={ e }^{ { x }_{ 1 }{ x }_{ 2 } }\\ \frac { \partial }{ \partial { x }_{ 1 } } =\frac { { x }_{ 3 } }{ 1+{ x }_{ 1 }^{ 2 }{ x }_{ 3 }^{ 2 } } \quad \frac { \partial }{ \partial { x }_{ 2 } } =0\quad \frac { \partial }{ \partial { x }_{ 3 } } =\frac { { x }_{ 1 } }{ 1+{ x }_{ 1 }^{ 2 }{ x }_{ 3 }^{ 2 } } \\ \Longrightarrow Df(x)=(\begin{matrix}{ { x }_{ 2 }{ x }_{ 3 }e }^{ { x }_{ 1 }{ x }_{ 2 } } & { { x }_{ 1 }{ x }_{ 3 }e }^{ { x }_{ 1 }{ x }_{ 2 } } & { e }^{ { x }_{ 1 }{ x }_{ 2 } } \\ \frac { { x }_{ 3 } }{ 1+{ x }_{ 1 }^{ 2 }{ x }_{ 3 }^{ 2 } } & 0 & \frac { { x }_{ 1 } }{ 1+{ x }_{ 1 }^{ 2 }{ x }_{ 3 }^{ 2 } } \end{matrix}) $$
d) $$ \frac { \partial }{ \partial { x }_{ 1 } } =(−{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 })−cos({ x }_{ 1 }+{ x }_{ 2 }))),\quad \frac { \partial }{ \partial { x }_{ 2 } } =(−{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 }))\quad \\ 1:\frac { { \partial }^{ 2 } }{ { \partial }^{ 2 }{ x }_{ 1 } } =−2{ e }^{ x }sin({ x }_{ 1 }+{ x }_{ 2 })\quad \quad 2:\frac { { \partial }^{ 2 } }{ { \partial }^{ 2 }{ x }_{ 2 } } =−{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 })+cos({ x }_{ 1 }+{ x }_{ 2 }))\\ 3:\frac { { \partial }^{ 2 } }{ { \partial }^{ 2 }{ x }_{ 1 } } =−{ e }^{ x }(sin({ x }_{ 1 }+{ x }_{ 2 })+cos({ x }_{ 1 }+{ x }_{ 2 }))\quad 4:\frac { { \partial }^{ 2 } }{ { \partial }^{ 2 }{ x }_{ 2 } } =−{ e }^{ x }cos({ x }_{ 2 }+{ x }_{ 1 }) $$