Mathematics and Statistics · Ch 3 — Differentiation

The Derivative — Recap and Standard Results

The derivative of a function measures its instantaneous rate of change — how fast the output $y = f(x)$ responds to a small change in the input $x$ . Geometrically it is the slope of the tangent to the graph of $f$ at a point. Formally, the derivative of $f$ at $x$ is defined as the limit of the average rate of change over a shrinking interval:

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

provided this limit exists. When it does, $f$ is said to be differentiable at $x$ . The Std XI treatment established this definition ("differentiation from first principles") and the derivatives of the basic functions; this Std XII chapter takes those as known and builds the powerful rules — product, quotient, chain, and the methods for composite, implicit, parametric, inverse and logarithmic forms — that let almost any function be differentiated quickly, without returning to the limit each time.

Notation. The derivative of $y = f(x)$ with respect to $x$ is written $\dfrac{dy}{dx}$ , $f'(x)$ , or $y'$ . All three mean the same thing. The instruction "differentiate" always means "find $\dfrac{dy}{dx}$ ."

Standard derivatives (used throughout). These results are quoted freely; they are the raw material every rule below combines.

Function $f(x)$	Derivative $f'(x)$
$c$ (constant)	$0$
$x^n$	$n\,x^{n-1}$
$\sqrt{x}=x^{1/2}$	$\dfrac{1}{2\sqrt{x}}$
$\dfrac{1}{x}=x^{-1}$	$-\dfrac{1}{x^2}$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\log x=\ln x$	$\dfrac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$

Note

"Differentiate" Means Find the Rate of Change — Recognise the Standard Form First

Before reaching for a rule, check whether the function is already one of the standard forms above (possibly after simplifying, e.g. rewriting $\sqrt{x}$ as $x^{1/2}$ or $\tfrac{1}{x^3}$ as $x^{-3}$ ). Rewriting into a power $x^n$ turns many awkward-looking terms into a one-step application of $\tfrac{d}{dx}x^n = n x^{n-1}$ .

Maharashtra's Std XII commerce Mathematics and Statistics syllabus draws on the same differential-calculus principles — the limit definition of the derivative, the standard-function results, and the rules developed below — that underpin calculus across Indian higher-secondary mathematics-and-statistics curricula, scoped here to algebraic, exponential, logarithmic, trigonometric and inverse functions rather than a full analysis course.

Definition 1Derivative (from first principles)

The derivative of $y=f(x)$ is $\dfrac{dy}{dx}=f'(x)=\displaystyle\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$ , when this limit exists — the instantaneous rate of change of $y$ with respect to $x$ , equal to the slope of the tangent to the graph at that point.

Definition 2Standard derivatives

Key results used throughout: $\tfrac{d}{dx}(x^n)=n x^{n-1}$ , $\tfrac{d}{dx}(e^x)=e^x$ , $\tfrac{d}{dx}(a^x)=a^x\ln a$ , $\tfrac{d}{dx}(\log x)=\tfrac1x$ , $\tfrac{d}{dx}(\sin x)=\cos x$ , $\tfrac{d}{dx}(\cos x)=-\sin x$ , and the derivative of any constant is $0$ .

← Previous chapterCh 2 — Matrices Next · 2 →Rules of Differentiation — Sum, Product and Quotient