Why is acceleration as a function of position or velocity equal to the derivative of velocity with respect to time? I understand that acceleration as a function of time is the derivative of velocity with respect to time, but how can that still be so if acceleration is a function of position or velocity?
 A: Acceleration is the derivative of velocity with respect to time, by definition.
It doesn't matter what factors affect it, that's still what it is. You could have acceleration as a function of the amount a spring is stretched, acceleration as a function of how much you press the gas pedal, acceleration of a sail boat as a function of how fast the wind is blowing, or whatever, and acceleration would still be defined as the derivative of velocity with respect to time.
A: If you are able to specify acceleration as a function of position or velocity, then it is still true that $$a=\frac{\text dv}{\text dt}=\frac{\text d^2x}{\text dt^2}$$
because this is just the definition of acceleration.

You encounter the position case in things like simple harmonic motion (like a mass on a spring). Then you can think of acceleration as a function of position where
$$a(x)=-\frac kmx$$
Solving the differential equation gives us what $x(t)$ is
$$x(t)=A\cos(\omega t+\phi)$$
where $\omega^2=k/m$, and $A$ and $\phi$ depend on initial conditions
You encounter the velocity case when considering drag forces. If a drag force is present that is proportional to the velocity, then we have 
$$a(v)=-\frac bmv$$
Solving the differential equation gives us what $v(t)$ is
$$v(t)=v_0\,e^{-b/m\,t}$$
where $v_0$ depends on the initial conditions.
You can also have both cases at the same time. For example, the damped harmonic oscillator:
$$a=-\frac kmx-\frac bmv$$
or really in many other scenarios. In general you could have a differential equation of the form 
$$a(x,v)=f(x)+g(v)$$
which you could write as a differential equation to solve for $x(t)$ as
$$\ddot x=f(x)+g(\dot x)$$

You can also have neither case. If your system is exhibiting some weird motion (like if I am just randomly moving a box back and forth across the ground), then there isn't even single-valued function $a(x)$ or $a(v)$ for this motion. However, there will always be $a(t)$.
Ultimately the time derivatives will always be the definition of acceleration, but this definition of acceleration does not depend on what we decide to express acceleration as a function of. 
