Mathematics and Statistics · Ch 9 — Differentiation

The Derivative as a Rate of Change

Differentiation measures how fast one quantity changes with respect to another. If $y = f(x)$ , the derivative of $y$ with respect to $x$ tells us the instantaneous rate of change of $y$ for a change in $x$ — geometrically, the slope of the tangent to the curve $y=f(x)$ at a point.

From average rate to instantaneous rate. Suppose $x$ changes from $x$ to $x+h$ ; then $y$ changes from $f(x)$ to $f(x+h)$ . The average rate of change over that interval is

\frac{\text{change in } y}{\text{change in } x} = \frac{f(x+h) - f(x)}{h}.

This is the slope of the secant line joining the two points. As we shrink the interval by letting $h \to 0$ , the secant swings into the tangent, and the average rate becomes the instantaneous rate — the derivative.

Notation. The derivative of $y=f(x)$ is written in several equivalent ways:

\frac{dy}{dx}, \qquad f'(x), \qquad y', \qquad \frac{d}{dx}\big[f(x)\big].

The value of the derivative at a particular point $x=a$ is written $f'(a)$ or $\left.\dfrac{dy}{dx}\right|_{x=a}$ , and it is the slope of the tangent at that point.

Why commerce students care. In commerce and economics the derivative is the language of marginal quantities: marginal cost is the derivative of total cost, marginal revenue the derivative of total revenue, and so on — each answers "how much does the total change for one more unit?" That is exactly an instantaneous rate of change.

Note

Derivative = Slope of the Tangent = Instantaneous Rate

These three phrases describe the same number: $f'(a)$ . A positive derivative means $y$ is increasing at that point, a negative derivative means it is decreasing, and a zero derivative means the tangent is horizontal (a momentary standstill). Keep this picture in mind — it makes every rule below intuitive.

Maharashtra's Std XI Commerce Mathematics and Statistics syllabus draws on the same standard principles of differential calculus taught across Indian higher-secondary and commerce-mathematics curricula — the derivative from first principles, the derivatives of standard functions, the rules of differentiation, the chain rule, and the second derivative developed in the rest of this chapter.

Definition 1Derivative

The derivative of $y=f(x)$ with respect to $x$ , written $\frac{dy}{dx}$ or $f'(x)$ , is the instantaneous rate of change of $y$ with respect to $x$ — equivalently, the slope of the tangent to the curve $y=f(x)$ at the point $x$ .

Definition 2Average vs. instantaneous rate of change

The average rate over $[x,\,x+h]$ is $\frac{f(x+h)-f(x)}{h}$ (the secant slope). Letting $h\to0$ turns it into the instantaneous rate $f'(x)$ (the tangent slope).

Definition 3Value of the derivative at a point

$f'(a)=\left.\frac{dy}{dx}\right|_{x=a}$ is the derivative evaluated at $x=a$ ; it gives the slope of the tangent to the curve at that specific point.

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