6.10. $f(x)=\sin \left(x^{2}\right)$.
Solution. This function is a complex two-link function: the first outer link is the sine, and the second inner link, the argument of the first link, is a quadratic function. The derivative of a composite function is the derivative of the first link with respect to its argument, keeping this argument, multiplied by the derivative of the second link with respect to $x$:

$$
(f(x))^{\prime}=\left(\sin \left(x^{2}\right)\right)^{\prime}=\cos \left(x^{2}\right) \cdot\left(x_{2}\right)^{\prime}=\cos \left(x^{2}\right) \cdot 2 x
$$

Incorrect solutions are often made as follows:

1. $(f(x))^{\prime}=\cos x \cdot 2 x$. Here, the argument $x^{2}$ of the first link is not preserved in the derivative of the first link.
2. $f^{\prime}(x)=\cos 2 x$. Here, the derivative of the second link is written in place of the argument of the first link.