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3.Chain Rule & Implicit Differentiation
Class 12 • Mathematics • NCERT
Chain Rule & Implicit Differentiation
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This lesson explains differentiation of composite functions using the chain rule and differentiation of equations where the dependent variable is not isolated, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand differentiation of composite functions.
- Apply the chain rule correctly.
- Learn implicit differentiation.
- Solve NCERT-based exam questions.
1. Composite Functions
A composite function is a function formed by applying one function to another.
If y = f(g(x)), then y is a composite function of x.
2. Chain Rule
If y = f(u) and u = g(x), then by chain rule:
dy/dx = dy/du × du/dx
3. Example of Chain Rule
Differentiate y = sin(3x²).
Let u = 3x²
y = sin u
y = sin u
dy/dx = cos u × du/dx
= cos(3x²) × 6x
4. Implicit Differentiation
When the dependent variable y is not expressed explicitly in terms of x, we use implicit differentiation.
Differentiate both sides with respect to x, treating y as a function of x.
5. Example of Implicit Differentiation
Differentiate: x² + y² = 25
Differentiating both sides:
2x + 2y (dy/dx) = 0
dy/dx = −x / y
6. Important NCERT Notes
• Chain rule is used for composite functions
• dy/dx must be written when differentiating y terms
• Implicit differentiation is common in board exams
• Used in applications of derivatives
• dy/dx must be written when differentiating y terms
• Implicit differentiation is common in board exams
• Used in applications of derivatives
Practice Questions (NCERT)
- State the chain rule.
- Differentiate y = cos(5x).
- Differentiate y = (2x + 1)³.
- Differentiate y = tan(x²).
- What is implicit differentiation?
- Differentiate x³ + y³ = 6xy.
- Why dy/dx is written in implicit differentiation?
- Is chain rule used for composite functions?
- Differentiate y = sin²x.
- Differentiate x² + y² = a².
✅ Show Answer Key
- dy/dx = dy/du × du/dx
- −5 sin(5x)
- 3(2x + 1)² × 2
- sec²(x²) × 2x
- Differentiation when y is not isolated
- dy/dx = (2y − x²) / (y² − 2x)
- Because y depends on x
- Yes
- 2 sin x cos x
- dy/dx = −x / y
© Aviate Learning – Chain Rule & Implicit Differentiation (NCERT Class 12)