Rates of Change & Related Rates

Emily Davis
7 min read
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Study Guide Overview
This study guide covers the meaning of a derivative in context, focusing on its interpretation as a rate of change. It explains how to determine the units of the derivative, interpret it in exam questions, and distinguish between a function representing an amount versus a rate. The guide includes worked examples involving real-world scenarios like depth of water and volume change, along with practice questions and a glossary of key terms like derivative, instantaneous rate of change, and slope of the tangent.
#Study Notes on Derivatives
#Table of Contents
- Introduction to Derivatives
- Meaning of a Derivative
- Worked Examples
- Practice Questions
- Glossary
- Summary and Key Takeaways
#Introduction to Derivatives
Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. They are essential for understanding rates of change and have numerous applications in various fields such as physics, engineering, and economics.
#Meaning of a Derivative
#What does the derivative mean?
- The derivative of a function represents the rate of change of that function.
- The rate of change describes how the dependent variable changes as the independent variable changes.
- The derivative, or the rate of change, is given by:
This means for every 1 unit increase in , increases by 3 units. In this case, the rate of change is constant (3) at every point on the graph of against .
- The derivative, or the rate of change, is:
This means changes at a rate of . Here, the rate of change depends on .
- At , the rate of change is:
- At , the rate of change is:
The rate of change at a particular point is the ...

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