Example 4. Find $f(0), f^{\prime}(0), f^{\prime \prime}(0), f^{\prime \prime \prime}(0), f^{\mathrm{IV}}(0)$, if $f(x)=\cos 2 x$.
Solution. We find the first, second, third, and fourth order derivatives:

$$
\begin{gathered}
f^{\prime}(x)=-2 \sin 2 x, f^{\prime \prime}(x)=-4 \cos 2 x, f^{\prime \prime \prime}(x)=8 \sin 2 x \\
f^{\mathrm{IV}}(x)=16 \cos 2 x
\end{gathered}
$$

By assigning the value of zero to \( x \), we find

$$
f(0)=1, f^{\prime}(0)=0, f^{\prime \prime}(0)=-4, f^{\prime \prime \prime}(0)=0, f^{\text {IV }}(0)=16
$$