Example 1. Find the derivative of the function $y=\sin 5 x$.
Solution. The argument of the sine here is not $x$, but $5 x$. This is a composite trigonometric function, which can be represented as:

$$
y=\sin z, z=5 x
$$

We have:

$$
y_{z}^{\prime}=(\sin z)_{z}^{\prime}=\cos z=\cos 5 x ; z_{x}^{\prime}=(5 x)_{x}^{\prime}=5
$$

Substituting the expressions for $y_{z}^{\prime}$ and $z_{x}^{\prime}$ into formula (2.11), we find:

$$
y_{x}^{\prime}=\cos 5 x \cdot 5,(\sin 5 x)^{\prime}=5 \cos 5 x
$$

The same result can be obtained directly, without explicitly introducing the intermediate argument $z=5 x$:

$$
(\sin 5 x)_{x}^{\prime}=(\sin 5 x)^{\prime} \cdot(5 x)_{x}^{\prime}=\cos 5 x \cdot 5=5 \cos 5 x
$$

Remark 1. In the last equalities, the prime denotes differentiation with respect to the intermediate argument $z=5 x$. The prime at the top and $x$ at the bottom denote differentiation with respect to $x$.

Remark 2. Here we used the fact that the derivative of the sine with respect to its argument is equal to the cosine of the same argument.