Answer
$\dfrac{dy}{dx}$ at $x=1$ is $\boxed{-8}$
Work Step by Step
We can expand the expression into a polynomial, which is relatively easy to take the derivative of. Note the identity $(x+y)(x-y) = x^2-y^2$.
$y=(1-x)(1+x)(1+x^2)(1+x^4)=(1-x^2)(1+x^2)(1+x^4)=(1-x^4)(1+x^4)=1-x^8$
Taking the derivative of this expression:
$\dfrac{dy}{dx} = -8x^7$
Evaluating $\dfrac{dy}{dx}$ at $x=1$ by plugging in $1$ gives $\boxed{-8}$.
