Algebraic solution to: Do the functions $y=\frac{1}{x}$ and $y=x^3$ ever have the same slope? The exercise doesn't specify how it must be answered, so I chose a graphical proof because I couldn't come up with an algebraic one. Sketching the graphs of $y=\frac{1}{x}$ and $y=x^3$, I noticed that $y=x^3$ always has a nonnegative slope, whereas $y=\frac{1}{x}$ always has a negative slope. Therefore these two functions never have the same slope.
However, I'm wondering if there's a algebraic way of showing this. I thought of differentiating each function and setting the values equal, but I think this would only prove that the two functions don't have the same slope at any particular x, and not that they don't have the same slopes anywhere over their domains.

Let $f(x)=x^3$ and $g(x)=x^{-1}$. We have $f'(x)=3x^2$ and $g'(x)=-x^{-2}$, now we do exactly what you said:
$Im(f')\cap Im(g')=\left\{t\in\mathbb{R}|t\geq0 \right\}\cap\left\{t\in\mathbb{R}|t<0 \right\}=\left\{t\in\mathbb{R}|t\geq0\wedge t<0 \right\}=\emptyset$
