Mathematical analysis of Marginal Cost Marginal cost is the cost associated with producing one more unit of output.         Mathematically speaking, marginal cost is equal to the change in total cost divided by the change in quantity.

$\ MC(q_{1},q_{2})=\frac{TC(q_{2})-TC(q_{1})}{q_{2}-q_{1}}$

Marginal cost can either be thought of as the cost of producing the last unit of output or the cost of producing the next unit of output. Because of this, it's sometimes helpful to think of marginal cost as the cost associated with going from one quantity of output to another, as shown by q1 and q2 in the equation below.

To get a true reading on marginal cost, q2 should be just one unit larger than q1.

That said, as we consider smaller and smaller changes in quantity, marginal cost converges to the derivative of total cost with respect to quantity.

$\ MC(q_{1},q_{2})=\frac{dTC}{dQ}$

What if we need to calculate marginal cost as we go from one output point to a much bigger one? 

For example, if the TC of producing 3 units of output is \$15 and the TC of producing 9 units of output is \$21, the marginal cost, simply put, is \$1.

Once we are considering a change in TC values due to a variation of quantity produced higher than 1 unity, are we applying the concept of average rate of change to measure the MC value?
I think you're getting things backwards. Marginal cost (at a certain quantity $q_0$) really only makes sense as the derivative of the total cost function with respect to quantity at point $q_0$:
$$
MC(q_0)=\frac{dTC(q_0)}{dq}.
$$
The interpretation of marginal cost as "the cost of producing the last unit of output or the cost of producing the next unit of output" is based on the (linear) approximation 
$$
\widehat{MC}^+(q_0)\approx\frac{TC(q_0+1)-TC(q_0)}{(q_0+1)-q_0}\quad\text{or}\quad
\widehat{MC}^-(q_0)\approx\frac{TC(q_0)-TC(q_0-1)}{q_0-(q_0-1)}
$$
This interpretation is usually introduced in undergraduate textbooks because of its intuitive appeal, not for its mathematical rigor. Imagine trying to explain the concept of marginal cost as the cost associated with an infinitesimal amount of change in quantity to a freshman who probably has no idea what "infinitesimal" means, let alone the concept of derivative. 

A first problem related to interpreting marginal cost as the cost associated with the previous or the next unit of output is that producing the two units (the previous and the next) may imply different costs. This is illustrated by the formulas $\widehat{MC}^+$ and $\widehat{MC}^-$. Suppose total cost is quadratic, i.e. $TC(q)=q^2$. Then it can be easily verified that $\widehat{MC}^+(q_0)\ne \widehat{MC}^-(q_0)$, for any admissible $q_0$. This is problematic because we want the marginal cost (at a particular $q_0$) to be unique. 

Second, and this relates to your question at the end, the concept of "unit" is somewhat arbitrary. And this arbitrariness makes measures such as $\widehat{MC}^+$ and $\widehat{MC}^-$ unstable as well. To illustrate, continue with the quadratic total cost example. When the unit of quantity is kg, the "marginal cost" at $1$kg--as you would refer to the cost associated with producing one extra unit--is $$\widehat{MC}^+(1)=2^2-1^2=3.$$
But what if we suddenly want to change the unit of quantity to gram (g)? Then the "marginal cost" at $1$kg, or $1000$g, would change as well:
$$
\widehat{MC}^+(1)=(1.001)^2-1^2=0.002001.
$$
To avoid such instability caused by changes in units of measurement, marginal cost is defined as a derivative, not as a difference. 

If you want to measure the change in total cost due to change in some arbitrary quantity, simply apply the first formula in your question. But keep in mind that this cost is not marginal cost as an economist would understand it.