Chain Rule Derivatives - Complete Tutorial with Examples

The power rule handles xⁿ. The product rule handles multiplication. But when one function is nested inside another — like sin(3x²) or e^(5x) — you need the chain rule.

The chain rule is the most-tested differentiation rule in calculus exams. Here's how to master it.

The Chain Rule Formula

d/dx[f(g(x))] = f'(g(x)) · g'(x)

Translation: differentiate the outer function (keeping the inner function intact), then multiply by the derivative of the inner function.

Think of it like peeling an onion: differentiate the outer layer first, then the next layer, and multiply the results.

Step-by-Step Examples

Example 1: sin(3x²)

1. Outer function: sin(u) → derivative = cos(u)
2. Inner function: 3x² → derivative = 6x
3. Chain rule: cos(3x²) · 6x = 6x·cos(3x²)

Example 2: e^(5x)

1. Outer: eᵘ → derivative = eᵘ
2. Inner: 5x → derivative = 5
3. Result: e^(5x) · 5 = 5e^(5x)

Example 3: ln(x² + 1)

1. Outer: ln(u) → derivative = 1/u
2. Inner: x² + 1 → derivative = 2x
3. Result: (1/(x²+1)) · 2x = 2x/(x²+1)

Example 4: (2x³ − 7)⁴

1. Outer: u⁴ → derivative = 4u³
2. Inner: 2x³ − 7 → derivative = 6x²
3. Result: 4(2x³−7)³ · 6x² = 24x²(2x³−7)³

The "Outside-In" Method

When you see a composite function, identify the outermost operation first:

| Function | Outermost Op | First Step | |----------|-------------|------------| | sin(3x) | sin | cos(3x) · (3) | | (x²+1)⁵ | power 5 | 5(x²+1)⁴ · (2x) | | √(4x+3) | square root | 1/(2√(4x+3)) · (4) | | e^(x²) | exponential | e^(x²) · (2x) |

The Trench Truth: The chain rule is the most frequently forgotten rule on exams. If your answer for d/dx(sin 3x) is "cos 3x" without the 3 multiplier, you forgot the chain rule. Always ask: "Is there a function inside another function?" If yes → chain rule.

Chain Rule + Other Rules Combined

Real exam problems combine multiple rules:

Example: x² · sin(3x)

This needs product rule + chain rule:

1. Product rule: (x²)' · sin(3x) + x² · (sin(3x))'
2. First part: 2x · sin(3x)
3. Second part needs chain rule: x² · cos(3x) · 3 = 3x²·cos(3x)
4. f'(x) = 2x·sin(3x) + 3x²·cos(3x)

Example: e^(x²) / (x+1)

This needs quotient rule + chain rule:

1. u = e^(x²), u' = 2x·e^(x²) (chain rule!)
2. v = x+1, v' = 1
3. f'(x) = [2x·e^(x²)·(x+1) − e^(x²)·1] / (x+1)²
4. f'(x) = e^(x²)·(2x²+2x−1) / (x+1)²

Common Chain Rule Mistakes

| Mistake | Wrong Answer | Correct Answer | |---------|-------------|----------------| | Forgetting inner derivative | d/dx(sin3x) = cos3x | d/dx(sin3x) = 3cos3x | | Chain rule on simple power | d/dx(x³) = 3x²·1 | d/dx(x³) = 3x² (chain rule gives same but unnecessary) | | Wrong inner derivative | d/dx(e^(3x²)) = 3e^(3x²) | d/dx(e^(3x²)) = 6x·e^(3x²) | | Multiple nesting errors | d/dx(sin(cosx)) = cos(sinx) | d/dx(sin(cosx)) = cos(cosx)·(−sinx) |

Practice Problems

Try these, then verify with our derivative calculator:

1. f(x) = cos(4x) → f'(x) = ?
2. f(x) = (3x − 1)⁷ → f'(x) = ?
3. f(x) = e^(2x²+1) → f'(x) = ?
4. f(x) = ln(5x³) → f'(x) = ?

Related: Integral Calculator · Quadratic Formula Calculator · Square Root Calculator