Is this writing incorrect?

Question

If we want to find $\frac{d}{dx}\cos x^2 $ then is this writing incorrect $\frac{d}{dx} \cos x^2= \frac{d}{dx^2}\cos x^2 \times \frac{d}{dx} x^2 $

Answer 1

No. That's correct. In general (chain rule):

$$\frac{d}{dt}F(g(t)) = \frac{d}{dt}g(t) \times \frac{d}{d(g(t))}F(g(t)) $$

Answer 2

No. This expression is not wrong. $cos(x^2)$ is just a composite function.

Derivative of such functions can be taken by using chain rule.