A derivative measures how a function changes. It's the mathematical way to describe rates of change.
The Big Idea
The derivative tells you how fast something is changing at a specific moment.
Think of it like this:
Speed is the derivative of position (how fast your location changes)
Acceleration is the derivative of speed (how fast your speed changes)
The slope of a curve at a point is the derivative
From Slope to Derivative
Remember the slope of a line between two points?
📚 Practice Problems
1Problem 1easy
❓ Question:
Use the definition of derivative to find f′(2) if .
Explain using:
📋 AP Calculus AB — Exam Format Guide
⏱ 3 hours 15 minutes📝 51 questions📊 4 sections
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⚠️ Common Mistakes: What is a Derivative?
Avoid these 4 frequent errors
🌍 Real-World Applications: What is a Derivative?
See how this math is used in the real world
📝 Worked Example: Related Rates — Expanding Circle
Problem:
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 10 cm?
Understanding the fundamental concept of derivatives
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Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
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What is a Derivative? is part of the AP Calculus AB course on Study Mondo, specifically in the Derivatives section. You can explore the full course for more related topics and practice resources.
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m=x2−x1y2−y1=ΔxΔy
The derivative is what happens when we make those points infinitely close together!
The Definition
The derivative of f(x) at x=a is:
f′(a)=limh→0hf(a+h)−f(a)
Let's break this down:
f(a+h): Function value slightly to the right
f(a): Function value at the point
f(a+h)−f(a): Change in y-values
h: Change in x-values (approaching 0)
The limit: Make h infinitesimally small
What This Fraction Means
hf(a+h)−f(a)
This is the average rate of change over the interval from a to a+h.
As h→0, we get the instantaneous rate of change (the derivative!)
Geometric Interpretation
The derivative at a point is the slope of the tangent line to the curve at that point.
Secant line: Connects two points on the curve
Tangent line: Touches the curve at exactly one point
As the two points get closer, the secant line → tangent line
Example: Computing from Definition
Find the derivative of f(x)=x2 at x=3 using the definition.
Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.