# HOW TO DERIVE MARGINAL COST FROM COST FUNCTION (CALCULUS)

Marginal cost is calculated as the change in total costs divided by the change in quantity.

However, the exact marginal cost is not always given. Rather, a cost function is usually given.

To derive marginal cost, we simply take the derivative of the cost function. Remember that marginal cost is the first derivative of the cost function

This is illustrated below.

Example 1
Given that the cost of producing shoes is $C(Q)=700+10Q+Q^2$. Find the marginal cost function of producing shoes.

Solution:
Total cost is given as, $C(Q)=700+10Q+Q^2$

Taking the derivative of the cost function, we have:
$C'(Q)=10+2Q$

Therefore,
$MC(Q)=10+2Q$

Thus the marginal cost function is $MC(Q)=10+2Q$

Example 2
A pen manufacturing firm cost function is $C(Q)=100-2Q+Q^2+10Q^3$. Find the marginal cost of producing the 70th pen.

Solution:
$C(Q)=100-2Q+Q^2+10Q^3$

Taking the derivatives of the cost function
$C'(Q)=-2+2Q+30Q^2$

Therefore,
$MC(Q)=-2+2Q+30Q^2$

By substitution
$MC(Q)=-2+2(70)+30(70)^2$
$MC=-2+140+30(4900)$
$MC=138+147000$
$MC=147138$

Example 3
Find the marginal cost of producing 10 units of goods if the cost function is $C(Q)=20+5Q+Q^2$

Solution:
$C(Q)=20+5Q+Q^2$

As usual, let's take the derivative
$C'(Q)=5+2Q$
$MC(Q)=5+2Q$

By substitution
$MC(10)=5+2(10)$
$=5+20$
$=25$

Example 4
A firm's cost function is given as $C(Q)=144+3Q+Q^2$. Find the quantity that has the lowest per-unit cost.

Solution:
per-unit cost is the same as the average total cost. To obtain the quantity that has the lowest per-unit cost, we equate marginal cost to average cost. Remember that the marginal cost curve intersects the average cost curve at the lowest point of the average cost curve.

$MC=ATC$

As usual, we take the derivative of the cost function
$C(Q)=144+3Q+Q^2$
$MC(Q)=3+2Q$

The average cost can easily be computed by divide the total cost by quantity
$ATC(Q)=\frac{144+3Q+Q^2}{Q}$

Equating MC to ATC
$3+2Q=\frac{144+3Q+Q^2}{Q}$

Cross-multiply
$Q(3+2Q)=144+3Q+Q^2$
$3Q+2Q^2=144+3Q+Q^2$
$3Q-3Q+2Q^2-Q^2=144$
$Q^2=144$
$\sqrt{Q^2}=\sqrt{144}$
$Q=12$

Therefore, the 12th units have the lowest per-unit cost.

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