Rate of Change and the Derivative — Introduction to Marginal Analysis

While the slope of a STRAIGHT line is constant everywhere, many important economic functions — total cost, total revenue — are CURVED, not straight, so their rate of change differs at different points. The derivative of a function, written $\dfrac{dy}{dx}$, is the precise mathematical tool for measuring the instantaneous rate of change of y with respect to x at any single point — effectively, the slope of the curve at that exact point.

Two simple derivative rules cover most of the functions economics uses at this level:

• Constant rule: the derivative of a constant term is always zero (a constant never changes, so its rate of change is nil).
• Power rule: for a term $ax^{n}$, the derivative is $anx^{n-1}$ — multiply by the original exponent, then reduce the exponent by 1.

Applying these rules to a total cost function $TC=Q^{2}+10Q+50$: the derivative of $Q^{2}$ is $2Q$ (power rule, $n=2$); the derivative of $10Q$ is $10$ (power rule with $n=1$, giving $10\times1\times Q^{0}=10$); the derivative of the constant $50$ is $0$. So $\dfrac{d(TC)}{dQ}=2Q+10$ — this IS the Marginal Cost function, confirming that Marginal Cost is simply the derivative of Total Cost with respect to output, and by the identical logic, Marginal Revenue is the derivative of Total Revenue with respect to output: $MR=\dfrac{d(TR)}{dQ}$. …