The Chain Rule

The chain rule is used when you need to differentiate a composite function — a function within a function. If $y = f(g(x))$ , then:

dy/dx = f'(g(x)) \cdot g'(x)

Differentiate the outer function, then multiply by the derivative of the inner function.

A useful way to remember the chain rule is: "derivative of the outside times derivative of the inside". For example, to differentiate $y = (3x + 1) 5$ , we let $u = 3x + 1$ , so $y = u 5$ .

Step 1: Differentiate the outer function: $dy/du = 5u 4$

Step 2: Differentiate the inner function: $du/dx = 3$

Result: $dy/dx = 5(3x + 1) 4 \times 3 = 15(3x + 1) 4$

The Product Rule

The product rule is used when differentiating the product of two functions. If $y = u(x) \cdot v(x)$ , then:

dy/dx = u'v + uv'

The derivative of the first times the second, plus the first times the derivative of the second.

For example, to differentiate $y = x 2 \cdot sin(x)$ , we let $u = x 2$ and $v = sin(x)$ .

u' = 2x, v' = cos(x)

dy/dx = 2x · sin(x) + x² · cos(x)

The Quotient Rule

The quotient rule is used to differentiate a function that is the ratio of two functions. If $y = u(x) / v(x)$ , then:

dy/dx = u'v - uv' v 2

"Low d-high minus high d-low, all over the square of what's below."

For example, to differentiate $y = (x + 1) / (x 2 - 3)$ , let $u = x + 1$ and $v = x 2 - 3$ .

u' = 1, v' = 2x

dy/dx = 1 · (x² − 3) − (x + 1) · 2x(x² − 3)²

= −x² − 2x − 3(x² − 3)²

Key Vocabulary

Chain Rule

A rule for differentiating composite functions: differentiate the outer function and multiply by the derivative of the inner function.

Product Rule

A rule for differentiating the product of two functions: u'v + uv'.

Quotient Rule

A rule for differentiating a ratio of two functions: (u'v − uv') / v².

Composite Function

A function formed by substituting one function into another, e.g. f(g(x)).

Worked Examples

1

Differentiate y = (2x³ − x)⁴ using the chain rule.

Step 1: Let u = 2x³ − x, so y = u⁴.

Step 2: dy/du = 4u³, and du/dx = 6x² − 1.

Answer: dy/dx = 4(2x³ − x)³(6x² − 1)

2

Differentiate y = x³ · e^x using the product rule.

Step 1: Let u = x³ and v = e^x. Then u' = 3x² and v' = e^x.

Step 2: Apply the product rule: dy/dx = u'v + uv' = 3x² · e^x + x³ · e^x.

Answer: dy/dx = e^x(3x² + x³) = x²e^x(3 + x)

3

Differentiate y = sin(x) / x² using the quotient rule.

Step 1: Let u = sin(x) and v = x². Then u' = cos(x) and v' = 2x.

Step 2: Apply the quotient rule: dy/dx = (u'v − uv') / v²

Step 3: = (cos(x) · x² − sin(x) · 2x) / x⁴

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

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Using the chain rule, find dy/dx if y = (5x + 2)³.

15(5x + 2)² 3(5x + 2)² 5(5x + 2)³ 15(5x + 2)³

Question 2

Using the product rule, differentiate y = x · ln(x).

1/x ln(x) + 1 x/ln(x) ln(x) − 1

Question 3

What is the derivative of y = e^3x?

e^3x 3xe^3x 3e^3x e³

Question 4

Using the quotient rule, differentiate y = x / e^x.

x / e^2x (1 + x) / e^x 1 / e^x (1 − x) / e^x

Question 5

Differentiate y = sin(x²) using the chain rule.

2x · cos(x²) 2x · sin(x²) cos(x²) x² · cos(x²)

Further Differentiation Techniques

The Chain Rule

The Product Rule

The Quotient Rule

Key Vocabulary

Chain Rule

Product Rule

Quotient Rule

Composite Function

Worked Examples

Differentiate y = (2x³ − x)⁴ using the chain rule.

Differentiate y = x³ · e^x using the product rule.

Differentiate y = sin(x) / x² using the quotient rule.

Knowledge Check

Key Concepts Summary

Further Differentiation Techniques

The Chain Rule

The Product Rule

The Quotient Rule

Key Vocabulary

Chain Rule

Product Rule

Quotient Rule

Composite Function

Worked Examples

Differentiate y = (2x3 − x)4 using the chain rule.

Differentiate y = x3 · ex using the product rule.

Differentiate y = sin(x) / x2 using the quotient rule.

Knowledge Check

Key Concepts Summary

Differentiate y = (2x³ − x)⁴ using the chain rule.

Differentiate y = x³ · e^x using the product rule.

Differentiate y = sin(x) / x² using the quotient rule.