Economics at Zero Marginal Cost: The Difference Quotient & The Derivation of Derivative

In comparative static, the problem under consideration is finding the rate of change i.e., the rate of change of the equilibrium value of an endogenous variable with respect to the change in a particular parameter or exogenous variable. The mathematical concept of derivative take an important significance in comparative statics. The concept of derivative is also important for optimisation.

Let us take a function $y=f(x)$ where $y$ changes in response to changes in variable $x$ .

THE DIFFERENCE QUOTIENT:

When variable $x$ changes from the value $x_{0}$ to a new value $x_{1}$ , the change is measured by the difference $x_{1}-x_{0}$ .

Using the symbol ∆ to denote change, we write

$\Delta x=x_{1}-x_{0}$

When $x$ changes from initial value x_o to a new value $x_{0}$ to a new value $x_{1}$ or $x_{0}+\Delta x$ , (since $\Delta x=x_{1}-x_{0}$ , therefore $x_{1}=x_{0}+\Delta x$ ), the new value of the function $y=f(x)$ changes from $f(x_{0})$ to $f(x_{0}+\Delta x)$ .

The change in $y$ per unit change in $x$ can be represented by the difference quotient.

$\frac{\Delta y}{\Delta x}=\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$

This quotient which measures the average rate of change of y , can be calculated if we know the initial value of $x$ or $x_{0}$ and the magnitude of change in $x$ or $\Delta x$ . That is $\frac{\Delta y}{\Delta x}$ is a function of $x_{0}$ and $\Delta x$ .

If $y=f(x)=6x^{2}-8$

Then,

$f(x_{0})=6(x_{0})^{2}-8$

$f(x_{0}+\Delta x)=6(x_{0}+\Delta x)^{2}-8$

Now, the difference quotient is:

$\\\frac{\Delta y}{\Delta x}=\frac{[6(x_{0}+\Delta x)^{2}-8]-[6(x_{0})^{2}-8]}{\Delta x}\\ \frac{\Delta y}{\Delta x}=\frac{6(x_{0}+\Delta x)^{2}-8-6(x_{0})^{2}+8}{\Delta x}\\ \frac{\Delta y}{\Delta x}=\frac{6(x_{0}^2+\Delta x^2+2x_{0}\Delta x)-6(x_{0})^{2}}{\Delta x}\\ \frac{\Delta y}{\Delta x}=\frac{6x_{0}^2+6\Delta x^2+12x_{0}\Delta x-6(x_{0})^{2}}{\Delta x}\\ \frac{\Delta y}{\Delta x}=\frac{6\Delta x^2+12x_{0}\Delta x}{\Delta x}\\ \frac{\Delta y}{\Delta x}=\frac{\Delta x(6\Delta x+12x_{0})}{\Delta x}\\ \frac{\Delta y}{\Delta x}=6\Delta x+12x_{0}\\$

which can be evaluated given $x_{0}$ and $\Delta x$ .

If $x_{0}=1$ and $\Delta x=2$ .

Then, the average rate of change of $y$ is:

$\frac{\Delta y}{\Delta x}=6(2)+12(1)=12+12=24$

DERIVATION OF THE DERIVATIVE:

In many cases, we are interested in the rate of change of $y$ when $\Delta x$ is very small. If the change in $x$ i.e., $\Delta x$ is very small, we can get an approximation of $\frac{\Delta y}{\Delta x}$ by dropping all the terms in the difference quotient involving the expression $\Delta x$ . The smaller the value of $\Delta x$ the closer the approximation to the true value of $\frac{\Delta y}{\Delta x}$ .

In $\frac{\Delta y}{\Delta x}=6\Delta x+12x_{0}\\$ we may take $12x_{0}$ as an approximation of $\frac{\Delta y}{\Delta x}$ .

In case of $\frac{\Delta y}{\Delta x}=6\Delta x+12x_{0}\\$ , as $\Delta x$ approaches zero (it gets closer and closer to, but never actually reaches zero) $6\Delta x+12x_{0}\\$ will approach the value $12x_{0}$ and hence $\frac{\Delta y}{\Delta x}$ will approach $12x_{0}$ also. Symbolically, $\frac{\Delta y}{\Delta x}\rightarrow 12x_{0}\ \ \ as\ \ \ \Delta x \rightarrow 0\\$ .

or by equation,

$\lim_{\Delta x\rightarrow 0}\frac{\Delta y} {\Delta x} =12x_{0}$

where lim is the limit of the function $\frac{\Delta y} {\Delta x}$ as $\Delta x$ approaches zero.

If as $\Delta x\rightarrow 0$ , the limit of the difference quotient does exists, the limit is called the derivative of the function $y=f(x)$ .

Hence, the derivative of a given function $y=f(x)$ can be written as:

$\frac{dy}{dx}=f'(x)=\lim_{\Delta x\rightarrow 0}\frac{\Delta y} {\Delta x}$

IMPORTANT POINTS:

A derivative is a function. The word derivative means a derived function. The original function $y=f(x)$ is a primitive function and derivative is another function derived from it.
Since the derivative is merely a limit of the difference quotient, which measures a rate of change of $y$ , the derivative must of necessity also be a measure of some rate of change.
Derivative functions are denoted in two ways:

Given a primitive function $y=f(x)$