Given the total cost function $C(Q)=Q^3-5Q^2+12Q+75$, write out a variable cost function. Find the derivative of the variable cost function and interpret the economic meaning of that derivative. Note: the derived variable cost function is 3Q^2-10Q+12
What is the difference between variable and fixed cost? Try separating your total cost into fixed and variable cost. To do this, think about what costs the firm faces even when they have no output (let quantity produced be zero.) For a bit of intuition, think of these costs perhaps as rents paid by a firm during the initial setup stage (time spent fitting a factory with production equipment). They have a certain amount of bills to pay even though they produce nothing.
After doing this it should be a straight forward thing to differentiate and interpret.
Let's take your example. We can see that we have a cost of 75 (let's assume American dollars) regardless of how much output we produce. So Q=0 still incurs a cost of $75. Then by definition, this is the fixed cost portion of your total cost function.
Then since any cost function is the sum of fixed and variable costs, we have that your variable cost function is $$C(Q)=Q^3-5Q^2+12Q$$
differentiating with respect to $Q$ yields: $$3Q^2-10Q+12$$
This is your marginal variable cost. Notice this is the same function achieved by simply differentiating the total cost function. This is because fixed costs are constants (and not functions of Q) and therefore disappear when C is differentiated with respect to Q.
Then we can just refer to the equation above as marginal cost
Marginal cost equations tell you how much your total cost rises when you decide to produce an additional unit of output. Notice here that making the production decision incurs a new sort of "fixed" cost.
So let's examine what happens when Q increases:
$$Q=1 \implies MC=5$$ $$Q=2 \implies MC=2$$ $$Q=3 \implies MC=9$$
and this is enough for us to tease out another important part of this function! Notice that marginal costs decrease as production increases over our initial range and then, at a certain point, marginal costs begin to increase as production continues to increase. This is a example of increasing marginal returns and decreasing marginal returns (respectively)
In general, you can picture a marginal cost curve as a differentiable nike symbol or a U where the right side of the U is drawn much higher than the left side(and where it is a bit slanted to the right This would tend to infinity as Q tends to infinity).
Alright - hope that cleared some things up. And to top it off, you can do a bit of reading about marginal cost.