State the chain rule for the derivative of a composition $t \mapsto f(\mathbf{X}(t))$, where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $\mathbf{X}: \mathbb{R} \rightarrow \mathbb{R}^{n}$ are smooth $.$

Consider parametrized curves given by

$$\mathbf{x}(t)=(x(t), y(t))=(a \cos t, a \sin t) .$$

Calculate the tangent vector $\frac{d \mathbf{x}}{d t}$ in terms of $x(t)$ and $y(t)$. Given that $u(x, y)$ is a smooth function in the upper half-plane $\left\{(x, y) \in \mathbb{R}^{2} \mid y>0\right\}$ satisfying

$$x \frac{\partial u}{\partial y}-y \frac{\partial u}{\partial x}=u$$

deduce that

$$\frac{d}{d t} u(x(t), y(t))=u(x(t), y(t))$$

If $u(1,1)=10$, find $u(-1,1)$.