Example 4. Find the derivative $y_{x}^{\prime}$ of the function $x=e^{t} \cos t$; $y=e^{t} \sin t$ at $t=0$.
Solution. The functions $x$ and $y$ have the following derivatives with respect to $t$:

$$
\begin{aligned}
& x_{i}^{\prime}=e^{t} \cos t-e^{t} \sin t=e^{t}(\cos t-\sin t) \\
& y_{t}^{\prime}=e^{t} \sin t+e^{t} \cos t=e^{t}(\sin t+\cos t)
\end{aligned}
$$

therefore

$$
y_{x}^{\prime}=\frac{\sin t+\cos t}{\cos t-\sin t}
$$

At $t=0$, we get $y_{x}^{\prime}=1$.