⭐ Sabse Important Points · एक नज़र में
- Limit: The value that a function approaches as the input approaches some value. Notation: lim_{x→a} f(x) = L.
- Derivative: f'(x) = lim_{h→0} [f(x+h) - f(x)]/h. Geometrically, it's the slope of the tangent.
- Continuity: A function f is continuous at x = a if lim_{x→a} f(x) = f(a).
- Chain Rule: d/dx [f(g(x))] = f'(g(x))·g'(x). Essential for composite functions.
- Differentiability implies continuity: If f is differentiable at a point, it is continuous there, but the converse is not always true.
- Maxima/Minima: Critical points where f'(x)=0 or undefined; use first or second derivative test to classify.
Introduction · परिचय
Differential Calculus is a fundamental topic in NDA Mathematics, focusing on derivatives, their properties, and applications. It tests understanding of limits, continuity, differentiability, and techniques of differentiation, along with geometric and physical interpretations. Mastery of this topic is crucial for solving problems related to rates of change, tangents and normals, maxima and minima, and curve sketching.
Limits and Continuity
The concept of a limit describes the behavior of a function near a point. Continuity at a point requires the limit to exist and equal the function value. These are foundational for differentiability.
📐 Formula / Rule Box · सूत्र याद रखो
Standard Limit
lim_{x→0} (sin x)/x = 1
Continuity Condition
lim_{x→a} f(x) = f(a)
⭐ Key Points · याद रखने वाली बातें
- Limit: The value that a function approaches as the input approaches some value. Notation: lim_{x→a} f(x) = L.
- Continuity: A function f is continuous at x = a if lim_{x→a} f(x) = f(a).
- Differentiability implies continuity: If f is differentiable at a point, it is continuous there, but the converse is not always true.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Evaluate lim_{x→0} (sin x)/x.
✅ Solution: 1
💡 Explanation: This is a standard limit. Using the squeeze theorem or L'Hôpital's rule (since it's 0/0 form), the limit is 1.
Differentiation Techniques and Applications
Differentiation gives the instantaneous rate of change. Key rules include product, quotient, and chain rules. Applications include finding tangents, normals, and extrema.
📐 Formula / Rule Box · सूत्र याद रखो
Product Rule
(uv)' = u'v + uv'
Quotient Rule
(u/v)' = (u'v - uv')/v^2
Chain Rule
dy/dx = dy/du · du/dx
Chain Rule
d/dx [f(g(x))] = f'(g(x))·g'(x). Essential for composite functions.
| First Derivative Test | Second Derivative Test |
|---|---|
| Sign change of f' from + to - gives local max | f'' < 0 gives local max |
| Sign change of f' from - to + gives local min | f'' > 0 gives local min |
| Works when f' changes sign | Works when f'' exists and nonzero |
⭐ Key Points · याद रखने वाली बातें
- Derivative: f'(x) = lim_{h→0} [f(x+h) - f(x)]/h. Geometrically, it's the slope of the tangent.
- Maxima/Minima: Critical points where f'(x)=0 or undefined; use first or second derivative test to classify.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the derivative of y = x^2 sin x.
✅ Solution: y' = 2x sin x + x^2 cos x
💡 Explanation: Apply the product rule: (uv)' = u'v + uv'. Here u = x^2, v = sin x, so u' = 2x, v' = cos x.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1If f(x) = |x|, then f is:
👀 Answer dekho
✅ Answer: Continuous at x=0 but not differentiable
💡 Explanation: The absolute value function is continuous everywhere, but at x=0, the left and right derivatives are -1 and 1, so derivative does not exist.
Q2Find dy/dx if y = ln(sin x).
👀 Answer dekho
✅ Answer: cot x
💡 Explanation: Using chain rule: dy/dx = (1/sin x)·cos x = cot x.
Q3The maximum value of f(x) = x^3 - 3x^2 + 6 on [0,3] is:
👀 Answer dekho
✅ Answer: 6
💡 Explanation: Find critical points: f'(x)=3x^2-6x=3x(x-2). Critical points at x=0,2. Evaluate f(0)=6, f(2)=2, f(3)=6. Maximum is 6.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Assuming differentiability implies continuity is true, but continuity does not imply differentiability (e.g., |x| at 0).
- Forgetting to apply chain rule for composite functions, leading to missing inner derivative.
- In maxima/minima problems, not checking endpoints of the interval, which can give absolute extrema.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Limit: lim_{x→a} f(x) exists if left and right limits are equal.
- Derivative: f'(x) = lim_{h→0} [f(x+h)-f(x)]/h; represents slope of tangent.
- Chain Rule: d/dx f(g(x)) = f'(g(x))·g'(x) – crucial for composite functions.
- Maxima/Minima: Find critical points where f'(x)=0 or undefined; use derivative tests to classify.