The Derivative — Definition from First Principles

Consider the graph of $y=f(x)$ and two points on it, $P=(x, f(x))$ and a nearby point $Q=(x+h, f(x+h))$, where $h$ is a small (positive or negative) change in $x$. The straight line through $P$ and $Q$ is a secant (chord), and its slope is

This ratio — change in $y$ over change in $x$ — is called the difference quotient. As $h \to 0$, the point $Q$ slides along the curve toward $P$, and the secant line rotates toward the tangent to the curve at $P$. The slope of that tangent — the instantaneous rate of change of $f$ at $x$ — is, by definition, the derivative of $f$ at $x$:

provided this limit exists (in which case $f$ is said to be differentiable at $x$). This way of obtaining the derivative — writing the difference quotient and taking the limit — is called finding the derivative from first principles.

The derivative is written in several equivalent notations: $f'(x)$, $\dfrac{dy}{dx}$ (when $y=f(x)$), or $Df(x)$. All three mean exactly the same limit above.

In a business setting, if $C(x)$ is the total cost of producing $x$ units, $C'(x)$ measures how fast cost changes per additional unit near $x$ — this specific interpretation (marginal cost, marginal revenue, and so on) is developed fully in the applications chapter that follows this one; here we build the tool itself. …