Chapter 12: Limits & Derivatives

1. Intuitive Idea of Derivatives

Calculus deals with the study of change. A Limit is the value that a function approaches as the input approaches some value.

Notation: \( \lim_{x \to a} f(x) = l \)

Limit Logic Lab

Observe \( f(x) = \frac{x^2 - 1}{x - 1} \) as \( x \to 1 \).

x (Input)	f(x) (Value)

As x gets closer to 1, f(x) gets closer to 2.

📌 Limits by Direct Substitution

If a function is continuous at \( x = a \), then:

\[ \lim_{x \to a} f(x) = f(a) \]

Examples:

\( \lim_{x \to 2} (x^2 + 3x) = 10 \)
\( \lim_{x \to 1} \frac{x^2 + 1}{x + 2} = \frac{2}{3} \)

CBSE Tip: These are quick 1–2 mark scoring questions.

2. Standard Limits Formulas

Limit Form	Value
\( \lim_{x \to a} \frac{x^n - a^n}{x - a} \)	\( n a^{n-1} \)
\( \lim_{x \to 0} \frac{\sin x}{x} \)	\( 1 \)
\( \lim_{x \to 0} \frac{1 - \cos x}{x} \)	\( 0 \)

L'Hôpital's Rule (Pro-Tip / Checking Only)

If \( \lim_{x \to a} \frac{f(x)}{g(x)} \) results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then:

\( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \)

Note: Very useful for quickly checking answers in MCQs or competitive exams, but use standard limits formulas for CBSE step-marking.

3. Derivative as Slope

The derivative \( f'(x) \) represents the slope of the tangent to the curve \( y = f(x) \) at any point \( x \).

Tangent Surfer

Surf the curve \( y = \frac{x^2}{10} \). Drag slider to change x.

Derivative from First Principle

The derivative of a function \( f(x) \) at point \( x \) is defined as:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Example:

Find derivative of \( f(x) = x^2 \).

Show Solution

\[ \frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h \]

Taking limit as \( h \to 0 \):

\[ f'(x) = 2x \]

CBSE Note: This is often asked as a 3–4 mark question.

4. Rules of Differentiation

If \( u \) and \( v \) are two functions of \( x \):

Sum & Difference Rule

\( \frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx} \)

Product Rule (Leibniz Rule)

\( \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \)

Example: Derivative of \( x\sin x \)

\( \frac{d}{dx}(x\sin x) = x\frac{d}{dx}(\sin x) + \sin x\frac{d}{dx}(x) = x\cos x + \sin x \)

Quotient Rule

\( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \)

Example: Derivative of \( \tan x = \frac{\sin x}{\cos x} \)

\( \frac{\cos x(\cos x) - \sin x(-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x \)

Derivatives of Basic Functions

Function	Derivative
\( x^n \)	\( nx^{n-1} \)
\( \sin x \)	\( \cos x \)
\( \cos x \)	\( -\sin x \)
Constant	0

Exam Tip: These are used directly in MCQs and short answers.

Power Rule Engine

Find derivative of \( f(x) = x^n \).

One-Page Revision Checklist

✔ Meaning of limit understood
✔ Limit notation clear
✔ Direct substitution rule revised
✔ Standard limits memorised
✔ Derivative as slope understood
✔ First principle formula revised
✔ Power rule & trig derivatives memorised

Self-Test:

Find \( \lim_{x \to 3} (x^2 - 1) \)
Find derivative of \( x^3 \)
What does derivative represent geometrically?

Concept Mastery Quiz

1. \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \) equals:

2. Derivative of a constant function is:

3. \( \frac{d}{dx}(\sin x) \) is:

4. Derivative of \( x^5 \) is:

5. Limit of \( \frac{\sin x}{x} \) as \( x \to 0 \) is: