Calculus: Early Transcendentals 8th Edition

This is not true. The counterexample is the function $f(x)=x^2$. Then we have $$\frac{dy}{dx}=\frac{d}{dx}(x^2)=2x\Rightarrow \left(\frac{dy}{dx}\right)^2=4x^2$$ and also $$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}(2x)=2.$$ We see that, obviously, $4x^2$ and $2$ are not identically equal (equal for every $x$).