Answer
The slope of $g^{-1}$ at the origin is $1/2$.
Work Step by Step
Here we do not know the formula for the function $y=g(x)$. But we know that it is differentiable, it has an inverse and its graph passes through the origin $(0,0)$ and has a slope of $2$ there.
Since the value of $dg/dx$ at a point $x=a$ is also the value of the slope at that point, the given information tells us that $$\left.\frac{dg}{dx}\right|_{x=0}=2$$
Now we need to find the slope of $g^{-1}$ at the origin. In other words, we need to find $dg^{-1}/dx$ at $x=0=f(0)$. According to Theorem 3,
$$\left.\frac{dg^{-1}}{dx}\right|_{x=0=f(0)}=\frac{1}{\left.\frac{dg}{dx}\right|_{x=0}}=\frac{1}{2}$$
Therefore, the slope of $g^{-1}$ at the origin is $1/2$.
