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Definition of the Derivative

Interactive secant-to-tangent limit convergence visualizing the formal definition of the derivative with dynamic dx collapsing.

WHAT IS A DERIVATIVE?

The derivative represents the instantaneous rate of change of a function. Geometrically, it is the slope of the tangent line.

LIMIT DEFINITION

The derivative is defined as .

CORE FORMULAS

Derivative definition

AP EXAM CONNECTION

Unit: Unit 2: Differentiation
Learning Objective: CHA-2.A

COMMON MISCONCEPTIONS

  • Confusing average rate of change with instantaneous rate of change.

KEY TAKEAWAYS

  • The derivative is a slope.

PRACTICE QUESTIONS

Q1 (QUANTITATIVE): Find the derivative of at using limits.

Show Answer & Explanation

Answer: 6

Explanation: Use the difference quotient and evaluate the limit.

DEEP DIVE: RELATED CONCEPTS

WHAT IS A LIMIT?

A limit describes the value that a function approaches as the input ($x$) gets closer and closer to ...

WHAT IS AN INTEGRAL?

The definite integral of a function $f(x)$ from $x=a$ to $x=b$ represents the net signed area betwee...

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