Exercise 3.6 Introduction ll Class 12 ll Chapter 3 ll Differentiation

Описание: 📘 *Mathematics Class 12 | Differentiation – Exercise 3.6 | Hyperbolic & Inverse Hyperbolic Functions*

Welcome to this detailed lecture on **Exercise 3.6: Introduction to Hyperbolic and Inverse Hyperbolic Functions**, from **Chapter 3: Differentiation (Class XII Mathematics)**.
This lecture is prepared for students of **Sindh Board, Federal Board, CBSE/ISC, Bangladesh Boards**, and **Cambridge A-Level**, focusing on conceptual understanding and differentiation techniques.

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🔹 *Topics Covered in Exercise 3.6 (Differentiation of Hyperbolic Functions)*

1️⃣ *Introduction to Hyperbolic Functions*

Definition of sinh x, cosh x, tanh x, coth x, sech x, and csch x.
Relation between hyperbolic and exponential functions.
Fundamental identities: cosh²x – sinh²x = 1, etc.

2️⃣ *Graphs and Properties*

Shape, domain, and range of major hyperbolic functions.
Comparison with trigonometric functions.

3️⃣ *Inverse Hyperbolic Functions*

Definitions of sinh⁻¹x, cosh⁻¹x, tanh⁻¹x, etc.
Principal values and domains.
Relationship with logarithmic forms.

4️⃣ *Differentiation of Hyperbolic Functions*

Derivatives of sinh x, cosh x, tanh x, coth x, sech x, csch x.
Derivatives of inverse hyperbolic functions.
Chain rule applications.
Simplifying results using hyperbolic identities.

5️⃣ *Standard Derivative Formulas*

( \frac{d}{dx}(\sinh x) = \cosh x )
( \frac{d}{dx}(\cosh x) = \sinh x )
( \frac{d}{dx}(\tanh x) = \text{sech}^2x )
( \frac{d}{dx}(\text{coth} x) = -\text{csch}^2x )
( \frac{d}{dx}(\text{sech} x) = -\text{sech}x\tanh x )
( \frac{d}{dx}(\text{csch} x) = -\text{csch}x\coth x )

6️⃣ *Important Results*

Derivatives of sinh⁻¹x, cosh⁻¹x, tanh⁻¹x, etc.
Using logarithmic forms for simplification.
Real-life application of hyperbolic functions in calculus and physics.

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🔹 *Why Exercise 3.6 is Important*
✅ Introduces a new family of functions beyond trigonometric ones.
✅ Builds strong base for higher calculus (Integration & Differential Equations).
✅ Frequently appears in board exams and university entrance tests.
✅ Strengthens algebraic manipulation and chain rule skills.

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🔹 *Keywords for Students (SEO Optimized)*

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Derivatives of sinh, cosh, tanh step-by-step
Inverse Hyperbolic Derivatives with Formulas
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Differentiation Exercise 3.6 Full Solutions
Online Math Lecture Differentiation Class 12

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🔹 *Motivation for Students*
✨ Hyperbolic functions look new — but behave just like trigonometric ones!
✨ Mastering them will make your next chapters (Integration & Differential Equations) easy.
✨ Remember: every new concept is a chance to grow your calculus power.
✨ “Don’t fear new symbols — they’re just old friends in disguise.”

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🔹 *Call to Action*
📌 Watch this lecture till the end to master all derivatives of hyperbolic and inverse hyperbolic functions.
📌 Subscribe to our channel for complete Class XII step-by-step tutorials.
📌 Like & Share with classmates to help them learn hyperbolic differentiation.
📌 Join online classes on Zoom/WhatsApp for board & entry test preparation (details in channel description).

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🔹 *Study Plan with This Lecture*
👉 Review definitions and exponential relationships first.
👉 Memorize standard derivatives — they reappear in Integration.
👉 Practice chain rule examples from Exercise 3.6.
👉 Attempt past board questions on hyperbolic differentiation.
👉 Revise identities regularly to avoid algebraic errors.

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🔹 *Upcoming Lectures in This Series*
📍 Exercise 3.7 — Logarithmic & Exponential Differentiation
📍 Exercise 3.8 — Parametric & Implicit Differentiation
📍 Review Exercise 3 — Complete Differentiation Revision

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🎯 *Final Note*
Exercise 3.6 connects algebraic, exponential, and calculus concepts into one beautiful topic — hyperbolic functions. Once you learn to differentiate them confidently, higher-level calculus becomes easier and more intuitive. Whether you’re in **Sindh Board, Federal Board, CBSE/ISC, Bangladesh, or Cambridge A-Level**, this lecture strengthens your base for all upcoming topics in differentiation and integration.

🚀 *Keep learning — practice is the bridge between confusion and clarity!*