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Year 11 Maths

The Product Rule

Learn how to differentiate the product of two functions using the formula d/dx[uv] = u'v + uv'.

What Is the Product Rule?

The product rule is used when you need to differentiate the product of two functions. If y = u × v, where both u and v are functions of x, you cannot simply differentiate each factor separately.

If y = uv, then:

dy/dx = u'v + uv'

or equivalently: d/dx[uv] = (du/dx)v + u(dv/dx)

In words: "the derivative of the first times the second, plus the first times the derivative of the second."

When to Use the Product Rule

Use the product rule whenever you have two separate functions multiplied together that cannot be easily expanded. Common examples include:

Step-by-Step Method

1

Identify the two functions: call them u and v.

2

Find u' (the derivative of u) and v' (the derivative of v).

3

Apply the formula: dy/dx = u'v + uv'.

4

Simplify by expanding and collecting like terms where possible.

Polynomial × Polynomial

y = x3(2x + 1)4

Polynomial × Exponential

y = x2ex

Polynomial × Trig

y = x sin(x)

Trig × Exponential

y = ex cos(x)

Combining Product Rule with Chain Rule

Often you will need to use both the product rule and the chain rule in the same problem. When differentiating each factor (u' or v'), you may need to apply the chain rule if that factor is itself a composite function.

Example: For y = x2(3x + 1)5, use the product rule with u = x2 and v = (3x + 1)5. When finding v', you need the chain rule: v' = 5(3x + 1)4 × 3 = 15(3x + 1)4.

Key Vocabulary

Product

The result of multiplying two or more quantities. In the product rule, y = uv is the product of functions u and v.

Product Rule

The differentiation rule stating d/dx[uv] = u'v + uv' for two differentiable functions u and v.

Factorise

To express a result as a product of factors. Often the final step in simplifying a product rule answer.

Derivative

The rate of change of a function, written as dy/dx, f'(x), or y'. Measures the gradient of the curve at any point.

Worked Examples

1

Differentiate y = x3 sin(x)

Step 1: Let u = x3 and v = sin(x).

Step 2: u' = 3x2 and v' = cos(x).

Step 3: dy/dx = u'v + uv' = 3x2 sin(x) + x3 cos(x).

Answer: dy/dx = 3x2 sin(x) + x3 cos(x) or x2(3 sin(x) + x cos(x)).

2

Differentiate y = (2x + 1)ex

Step 1: Let u = 2x + 1 and v = ex.

Step 2: u' = 2 and v' = ex.

Step 3: dy/dx = 2 × ex + (2x + 1) × ex = ex(2 + 2x + 1).

Answer: dy/dx = ex(2x + 3).

3

Differentiate y = x2(x - 4)3

Step 1: Let u = x2 and v = (x - 4)3.

Step 2: u' = 2x and v' = 3(x - 4)2 (chain rule).

Step 3: dy/dx = 2x(x - 4)3 + x2 × 3(x - 4)2.

Step 4: Factor out x(x - 4)2: dy/dx = x(x - 4)2[2(x - 4) + 3x] = x(x - 4)2(5x - 8).

Answer: dy/dx = x(x - 4)2(5x - 8).

Knowledge Check

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

Question 1

If y = x × ex, what is dy/dx?

Question 2

Differentiate y = x2 cos(x). What is dy/dx?

Question 3

In the product rule formula dy/dx = u'v + uv', what does u' represent?

Question 4

Differentiate y = (3x + 2)(x2 - 1). What is dy/dx?

Question 5

Differentiate y = x e2x. What is dy/dx?

Key Concepts Summary

The Chain Rule The Quotient Rule