Statistics · Ch 9 — Differentiation

Meaning of the Derivative and Rate of Change

Differentiation is the branch of mathematics that studies how one quantity changes in response to a change in another — its rate of change. For a Gujarat Board (GSHSEB) Std 12 Commerce Statistics student, this idea matters directly: cost, revenue, profit and demand are all quantities that change as output, price, or time changes, and the derivative is the precise tool that measures exactly how fast.

Average Rate of Change

Let $y = f(x)$ . If $x$ changes from $x$ to $x + \Delta x$ (where $\Delta x$ is a small change), $y$ changes correspondingly by $\Delta y = f(x+\Delta x) - f(x)$ . The ratio

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

is called the average rate of change of $y$ with respect to $x$ over that interval.

The Derivative — Instantaneous Rate of Change

As $\Delta x$ is made smaller and smaller, approaching zero, the average rate of change approaches a fixed limiting value called the derivative of $f(x)$ with respect to $x$ , written $f'(x)$ or $\dfrac{dy}{dx}$ :

f'(x) = \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

This is called the instantaneous rate of change of $y$ with respect to $x$ at the point $x$ , and the process of finding it is called differentiation. Geometrically, $f'(x)$ gives the slope of the tangent line to the curve $y = f(x)$ at that point.

Notation: the derivative of $y$ with respect to $x$ is written interchangeably as $\dfrac{dy}{dx}$ , $f'(x)$ , $y'$ , or $D_x y$ — all mean exactly the same thing.

Finding a Derivative from First Principles

Finding $f'(x)$ directly from the limit definition above is called differentiating from first principles. For example, to differentiate $f(x) = x^2$ :

f'(x) = \lim_{\Delta x \to 0} \frac{(x+\Delta x)^2 - x^2}{\Delta x} = \lim_{\Delta x \to 0} \frac{x^2 + 2x\Delta x + (\Delta x)^2 - x^2}{\Delta x} = \lim_{\Delta x \to 0} (2x + \Delta x) = 2x

So $\dfrac{d}{dx}(x^2) = 2x$ . In practice, once the standard rules of this chapter are known, first-principles differentiation is rarely needed for routine problems — but understanding it is essential, because every rule in this chapter is itself derived from this same limit definition, and Gujarat Board (GSHSEB) Std 12 Commerce Statistics examinations regularly ask for at least one derivative found from first principles to test this understanding directly.

Definition 1Derivative ($f'(x)$ or $dy/dx$)

The limit of the average rate of change of $y$ with respect to $x$ as the change in $x$ approaches zero; equals the slope of the tangent to the curve $y=f(x)$ at that point.

Definition 2Differentiation

The process of finding the derivative of a function.

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