Final answer:
The inverse of a function being differentiable relies on the differentiability of the original function. If the original function is differentiable, then its inverse will be differentiable as well.
Step-by-step explanation:
The inverse of a function being differentiable relies on the differentiability of the original function. If the original function is differentiable, then its inverse will be differentiable as well. However, if the original function is not differentiable at a certain point, then its inverse will also not be differentiable at that point.
This can be understood by considering the graph of the original function and its inverse. If the original function has a sharp corner or vertical tangent at a certain point, then its inverse will also have a sharp corner or vertical tangent at that point, making it non-differentiable.
For example, consider the function f(x) = x^3. This function is differentiable everywhere, so its inverse will also be differentiable everywhere. However, if we consider the function g(x) = |x|, which has a sharp corner at x = 0, then its inverse will also have a sharp corner at x = 0, making it non-differentiable at that point.