Asymptotics of 2Ai(x)Ai'(x)

I'm looking for the asymptotics of the derivative of Airy squared $2Ai(x)Ai'(x)$ as $x\to \infty$. I know that $A i^2(x) \sim \frac{1}{4} \pi^{-1}(x)^{-1 / 2} e^{-\frac{4}{3}(x)^{3 / 2}}$. Would it be correct simply take the derivative of both sides to obtain?

$$\frac{d}{dx}Ai^2(x) = 2Ai(x)Ai'(x)\sim -\frac{e^{-\left(4 x^{3 / 2}\right) / 3}\left(4 x^{3 / 2}+1\right)}{8 \pi x^{3 / 2}}$$

Taking this one step further, using that the Airy equation satisfies $f''(x)=xf(x)$, and taking the derivative of $2Ai(x)Ai'(x)$, would it be correct to say that

$$\frac{d^2}{dx^2}Ai(x)=2(Ai'(x))^2+2Ai(x)Ai''(x)=2(Ai'(x))^2+2xAi^2(x) \sim\frac{e^{-\left(4 x^{3 / 2}\right) / 3}\left(4 x^{3 / 2}+16 x^3+3\right)}{16 \pi x^{5 / 2}}? $$

Thank you!