Why when traveling a curved path does acceleration go inwards?

[proto3210]

Hello, I am having some trouble wrapping my mind around this concept. In the beginning of calculus I was taught that the derivative is relationship of change of one variable with respect to another variable, and that a good way to visualize this is the tangent of a curve at a point. My physics textbook notes that when traveling a curved path the instantaneous acceleration is pointed towards the concave side of the path. I do not quite understand why?

To find instantaneous velocity, even around a curved path it is the tangent line of the curve at a point. Why is acceleration different?

 

[Orodruin]

Because acceleration is the change in velocity. How does the velocity change when you make a turn?