Skip to main content
MathidiaMathematics Academy
Advanced differentiation
Lesson 26 of 75
Introduction

Product rule

Objective. Differentiate products and simplify strategically.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The product rule differentiates a function made of two factors multiplied together. If y = uv, where u and v are both functions of x, then dy/dx = u(dv/dx) + v(du/dx). You differentiate each factor in turn while keeping the other one fixed, then add the two results. For example, if y = x²sin x, take u = x² and v = sin x, so dy/dx = x²cos x + 2x sin x. Always identify u and v clearly before differentiating each one separately.

Worked example

Differentiate y = x³ln x using the product rule.

y = mx + c
Coordinate grid with the line y = mx + c
  1. 1Let u = x³ and v = ln x, so du/dx = 3x² and dv/dx = 1/x
  2. 2Apply dy/dx = u(dv/dx) + v(du/dx)
  3. 3dy/dx = x³(1/x) + ln x(3x²)
  4. 4Simplify to dy/dx = x² + 3x²ln x

Answer: dy/dx = x² + 3x²ln x

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. A common error is multiplying the two derivatives together, writing dy/dx = (du/dx)(dv/dx). This ignores the sum structure entirely; always form u(dv/dx) + v(du/dx) and never multiply the derivatives.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard