Chain Rule Practice Problems with Step-by-Step Solutions

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By Bikash Roy

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Bikash Roy

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· June 1, 2026

How to Use These Problems

Work through each problem before reading the solution. The problems increase in difficulty — Problems 1–3 are basic one-layer compositions, Problems 4–6 use combinations with product or quotient rule, and Problems 7–8 are nested (multiple chain rule applications).

For each problem, the steps follow the standard approach from the Step-by-Step Chain Rule Guide: identify outer and inner functions, find dy/du and du/dx, then multiply. Use the Chain Rule Calculator to verify any result.

Problem 1 — Basic Power (Beginner)

Find the derivative of y = (2x + 5)^4

Step 1 — Identify structure:

• Outer function: f(u) = u^4
• Inner function: g(x) = 2x + 5

Step 2 — Differentiate outer (keep inner inside):

• dy/du = 4u^3 = 4(2x + 5)^3

Step 3 — Differentiate inner:

• du/dx = 2

Step 4 — Multiply:

dy/dx = 4(2x + 5)^3 × 2 = 8(2x + 5)^3

Answer: dy/dx = 8(2x + 5)^3

Problem 2 — Trig with Polynomial Inner (Beginner)

Find the derivative of y = sin(x³)

Step 1 — Identify structure:

• Outer: f(u) = sin(u)
• Inner: g(x) = x³

Step 2 — Differentiate outer:

• dy/du = cos(u) = cos(x³)

Step 3 — Differentiate inner:

• du/dx = 3x²

Step 4 — Multiply:

dy/dx = cos(x³) × 3x² = 3x² cos(x³)

Answer: dy/dx = 3x² cos(x³)

Problem 3 — Exponential with Linear Inner (Beginner)

Find the derivative of y = e^(4x − 1)

Step 1 — Identify structure:

• Outer: f(u) = e^u
• Inner: g(x) = 4x − 1

Step 2 — Differentiate outer:

• dy/du = e^u = e^(4x − 1)

Step 3 — Differentiate inner:

• du/dx = 4

Step 4 — Multiply:

dy/dx = e^(4x − 1) × 4 = 4e^(4x − 1)

Answer: dy/dx = 4e^(4x − 1)

Problem 4 — Logarithm with Trig Inner (Intermediate)

Find the derivative of y = ln(cos x)

Step 1 — Identify structure:

• Outer: f(u) = ln(u)
• Inner: g(x) = cos x

Step 2 — Differentiate outer:

• dy/du = 1/u = 1/cos(x)

Step 3 — Differentiate inner:

• du/dx = −sin x

Step 4 — Multiply:

dy/dx = (1/cos x) × (−sin x) = −sin(x)/cos(x) = −tan(x)

Answer: dy/dx = −tan(x)

This is a standard result: d/dx[ln(cos x)] = −tan(x).

Problem 5 — Product Rule + Chain Rule (Intermediate)

Find the derivative of y = x² · e^(3x)

This is a product of x² and e^(3x). Apply product rule first, then chain rule on e^(3x).

Product rule: d/dx[f·g] = f’·g + f·g’

• f = x², f’ = 2x
• g = e^(3x), g’ = chain rule → e^(3x) · 3 = 3e^(3x)

Combine:

dy/dx = (2x)·e^(3x) + x²·(3e^(3x))
      = 2x·e^(3x) + 3x²·e^(3x)
      = e^(3x)(2x + 3x²)
      = xe^(3x)(2 + 3x)

Answer: dy/dx = xe^(3x)(2 + 3x)

Problem 6 — Quotient Rule + Chain Rule (Intermediate)

Find the derivative of y = sin(2x) / x²

Apply quotient rule: d/dx[f/g] = (f’g − fg’) / g²

• f = sin(2x) → f’ = cos(2x)·2 = 2cos(2x) [chain rule on sin(2x)]
• g = x² → g’ = 2x

Combine:

dy/dx = [2cos(2x)·x² − sin(2x)·2x] / x⁴
      = 2x[x·cos(2x) − sin(2x)] / x⁴
      = 2[x·cos(2x) − sin(2x)] / x³

Answer: dy/dx = 2[x·cos(2x) − sin(2x)] / x³

Problem 7 — Double Nested Composition (Advanced)

Find the derivative of y = e^(sin(x²))

Three layers: e → sin → x². Work from outside in.

Step 1 — Outermost layer (e^u):

• d/dx[e^u] = e^u · du/dx where u = sin(x²)
• Result so far: e^(sin(x²)) · d/dx[sin(x²)]

Step 2 — Middle layer (sin(v)) where v = x²:

• d/dx[sin(x²)] = cos(x²) · d/dx[x²] = cos(x²) · 2x

Step 3 — Combine all three multiplications:

dy/dx = e^(sin(x²)) · cos(x²) · 2x
      = 2x · cos(x²) · e^(sin(x²))

Answer: dy/dx = 2x·cos(x²)·e^(sin(x²))

Problem 8 — Chain Rule with Radical and Trig (Advanced)

Find the derivative of y = √(1 + sin²(x))

Rewrite as y = (1 + sin²(x))^(1/2). Three layers: sqrt → sin² → sin → x.

Actually this is two nested compositions:

• Outer: (·)^(1/2) applied to [1 + sin²(x)]
• Inner: 1 + sin²(x) which itself needs chain rule for sin²(x)

Step 1 — Apply power rule on outer:

• dy/dx = (1/2)(1 + sin²(x))^(−1/2) · d/dx[1 + sin²(x)]

Step 2 — Differentiate 1 + sin²(x):

• d/dx[1] = 0
• d/dx[sin²(x)] = d/dx[(sin x)^2] = 2·sin(x)·cos(x) = sin(2x)

Step 3 — Combine:

dy/dx = (1/2)(1 + sin²(x))^(−1/2) · sin(2x)
      = sin(2x) / (2√(1 + sin²(x)))

Answer: dy/dx = sin(2x) / (2√(1 + sin²(x)))

Key Patterns from These Problems

For the complete reference of all chain rule patterns organized by function type, see the Chain Rule Patterns Reference. For chain rule applied in implicit differentiation context, see Chain Rule in Implicit Differentiation.