The derivation of average fixed cost, curve, average variable cost curve, average total cost curve and marginal cost curve may be explained as follows:-
Average Fixed Cost (AFC)
The average fixed cost can be obtained by dividing total fixed cost by the quantity of output. It is expressed as,
As the output increases, average fixed cost [AFC] goes on declining. For example, when output is one unit the average fixed cost is Rs.60, and when output increases to 5 units the average fixed cost declines to Rs. 12.
Average Variable Cost (AVC)
The variable cost per unit of output is called average variable cost. It is expressed as, 
The average variable cost (AVC) decreases in the beginning when output increases and begins to increases after normal capacity is reached. For example, up to output 3 units, average variable cost declines from Rs. 20 to Rs .15, after that increases to Rs.27 when output increases to 5 units.
Average Total Cost (ATC)
The average total cost is, in fact, only average cost. It is the cost per unit of output. It is expressed as,
Or ATC = AFC + AVC
The average total cost is obtained by dividing total cost by quantity of output. Alternatively, it may be obtained by adding average fixed cost and average variable cost.
Average total cost (ARC) is the sum of average fixed cost and average variable cost. Average total cost also declines in the beginning as the output increases, and begins to increases after normal capacity is reached. For example, average total cost declines from Rs.80 to Rs. 35 when output is 4 units. After that it increases to Rs. 39 when output is 5 units.
Marginal Cost (MC)
The marginal cost is the additional made to total cost when one more unit is produced. It is calculated as
The marginal cost of 2 units of output is obtained by subtracting the total cost of one unit of output from two units of output produced.
The marginal cost is the change in total cost due to the change in output. Hence, it is also calculated as, 
The marginal cost is independent of the fixed cost because the fixed cost does not change with output. The total cost changes due to change in variable cost. Hence, the marginal cost is also calculated as,
Marginal cost also declines in the beginning and after a point is reached it begins to increase. For example, when output is one unit marginal cost is Rs. 20 and begins to decline after that. Marginal cost increases to Rs.35 when output is 4 units and begins to increase after that.
The derivation of short-run average and marginal cost curves can be explained by the help of figure.
In figure, AFC curve represents average fixed cost curve. It slopes downwards to the right. As the output increases, the ratio of fixed cost to output declines. It is because the fixed cost is a fixed quantity. The AFC curve is a rectangular hyperbola. The AFC approaches both the axes but does not touch them.
The AVC curve in the figure represents average variable cost. It slopes downward in the beginning, reaches the minimum point and rises upward thereafter. In general, the AVC curve is ‘U’ shaped. In general, the average variable cost declines up to the normal capacity. The decline in AVC in the beginning is due to the operation of law of diminishing returns. After normal capacity, the AVC increases rapidly.
In figure, ATC curve is the average total cost curve. In the beginning both AFC and AVC fall so ATC falls rapidly but even after AVC rises upward. AFC continues to fall rapidly. So ATC continue to fall. But if production is further increased, AVC increases rapidly. Hence, ATC rises after a point. In this way, the ATC first falls, reaches minimum point and rises thereafter. The ATC is almost ‘U’ shaped.
In figure, MC is the marginal cost curve. As shown in the figure, the marginal cost also first falls, reaches the minimum point and increases thereafter. Hence, the MC curve also first slopes downward, reaches the minimum point and rises thereafter. The upward-sloping MC curve intersects AVC and ATC curves in their minimum points from below.