Consider all lines that are simultaneously tangent to the parabolas $y = x^2 + 1$ and $y = −x^2 − 1$. What's the product of these lines' slopes?
Taking the derivative, I found two slopes, whose product is $-4$. The correct solution, however, is $-1$.
The tangent line to a function $f$ at $x=x_0$ has equation $$y=f'(x_0)(x-x_0)+f(x_0)$$
Therefore the tangents to the first parabola are given by $$y=2x_0x-x_0^2+1$$ and the tangents to the second one by $$y=-2x_1x+x_1^2-1$$
The equations match if and only if $x_0=\pm 1$ and $x_1=\mp1$.
Therefore the product of the slopes is $2\times (-2)=-4.$