Chain rule: Class 12 practice
To differentiate a function of a function, differentiate the outside (keeping the inside) and multiply by the derivative of the inside: d/dx f(g(x)) = f′(g(x)) g′(x).. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Standard derivatives
The idea behind chain rule
To differentiate a function of a function, differentiate the outside (keeping the inside) and multiply by the derivative of the inside: d/dx f(g(x)) = f′(g(x)) g′(x).
Step-by-step method
- Spot the inner function.
- Differentiate the outer function.
- Multiply by the inner derivative.
- Simplify.
Worked example 1 (Basic)
Find dy/dx at x = 0 for y = (3x + 4)^4.
- dy/dx = 4(3x + 4)^3 × 3
- At x = 0: 768
- Answer: 768
Worked example 2 (Standard)
Differentiate y = sin(5x² + 2).
- Outer: cos(…), inner derivative 10x
- Answer: 10x cos(5x² + 2)
Worked example 3 (Challenge)
Differentiate y = e^(6x + 3) and find dy/dx at x = 0.
- dy/dx = 6e^(6x + 3)
- At 0: 6e^3 ≈ 120.5132
- Answer: 6e^(6x + 3); at 0 ≈ 120.5132
Common mistakes and misconceptions
- Forgetting the inner derivative
- Differentiating the inside twice
Practice questions
- (Basic) Find dy/dx at x = 1 for y = (4x + 2)^5.
- (Basic) Find dy/dx at x = 1 for y = (3x − 5)^5.
- (Standard) Differentiate y = sin(6x² − 5).
- (Standard) Differentiate y = sin(3x² + 2).
- (Challenge) Differentiate y = e^(4x − 2) and find dy/dx at x = 0.
- (Challenge) Differentiate y = e^(6x − 4) and find dy/dx at x = 0.
Answer key with full working
- Q1: dy/dx = 5(4x + 2)^4 × 4 → At x = 1: 25920 → Answer: 25920
- Q2: dy/dx = 5(3x − 5)^4 × 3 → At x = 1: 240 → Answer: 240
- Q3: Outer: cos(…), inner derivative 12x → Answer: 12x cos(6x² − 5)
- Q4: Outer: cos(…), inner derivative 6x → Answer: 6x cos(3x² + 2)
- Q5: dy/dx = 4e^(4x − 2) → At 0: 4e^-2 ≈ 0.5413 → Answer: 4e^(4x − 2); at 0 ≈ 0.5413
- Q6: dy/dx = 6e^(6x − 4) → At 0: 6e^-4 ≈ 0.1099 → Answer: 6e^(6x − 4); at 0 ≈ 0.1099
Syllabus note
This skill is listed under "Continuity and Differentiability" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is chain rule in?
"Continuity and Differentiability", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.