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MathematicsHigh school

Derivatives from scratch

What a derivative is, how to compute one, and what it's good for - three short lessons with fully worked examples.

3 lessons · about 42 min

Start the course

After this course you will

  • Explain what a derivative is and what it tells you about the graph of a function
  • Compute the derivative of simple functions from the definition
  • Use the sum, product, and quotient rules
  • Find the derivative of a composite function

About the course

Derivatives are a key topic in advanced final-year high-school math and the foundation of first-year university calculus in many majors. This course takes you from intuition to confident computation: every lesson ends with examples solved step by step, and at the end you’ll find exercises to work through on your own.

What you need to get started

  • linear and quadratic functions and their graphs,
  • the laws of exponents and special products such as (a+b)2(a + b)^2,
  • an intuitive sense of limits - it’s enough to know what “hh approaches zero” means.

Sources

We checked the facts on this page against these sources. All of them may be reused freely: they carry an open licence or are not covered by copyright.

  1. Bressoud, Ghedamsi, Martinez-Luaces and Törner - Teaching and Learning of Calculus (ICME-13 Topical Surveys, Springer, 2016), 2.4 calculus instruction in Germany, the United States and Uruguay (taught at the end of secondary school, a first university course for many majors, derivatives as a central concept) · CC BY 4.0
  2. Biza, González-Martín and Pinto - 'Scaffolding' or 'Filtering': A Review of Studies on the Diverse Roles of Calculus Courses, International Journal of Research in Undergraduate Mathematics Education 8, 2022 (calculus at upper secondary level and in the first year of many STEM programs) · CC BY 4.0
  3. Jameson, Machaba and Fasinu - Misconceptions and Errors Among Grade 12 Students When Learning Differentiation Rules, Mathematics Education Journal 8(2), 2024 (differentiation in the final school year in South Africa) · CC BY-SA 4.0
  4. Active Calculus (Boelkins) - 1.3 The derivative of a function at a point · CC BY-SA 4.0
  5. Dale Hoffman - Contemporary Calculus, Chapter 2: The Derivative (definition, differentiation rules, chain rule) (PDF) · CC BY 3.0 US
  6. Calaway, Hoffman and Lippman - Applied Calculus, Chapter 2: The Derivative (definition, power, sum, product, quotient and chain rules) (PDF) · CC BY 3.0 US

Lessons

  1. 1What is a derivative?12 min
  2. 2Differentiation formulas and rules15 min
  3. 3The chain rule: derivatives of composite functions15 min

See it animated

MathematicsHigh school

What is a derivative? Explained step by step

What a derivative is, where it comes from and how to work it out - with an animation of a secant turning into a tangent, worked examples and a quiz.

10 min read

Exercises for this course

MathematicsHigh schoolMedium

Product rule and chain rule together

Differentiate x√(x² + 9) using the product rule and the chain rule, then work out the value of the derivative at x = 4.

MathematicsHigh schoolHard

Local extrema of a cubic function

Use the derivative to find the intervals where a function is increasing or decreasing and its local extrema.

MathematicsHigh schoolHard

Open box of maximum volume

Use the derivative to find what size of square to cut from each corner of a 30 cm × 48 cm sheet so that an open box has the largest volume.