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Unit: Derivatives introduction

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The derivative: An intuitive introduction

Learn
Newton, Leibniz, and Usain Bolt
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Derivative as a concept
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Derivative notation review
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Derivative as slope of curve
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Derivative & the direction of a function
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Practice
Derivative as slope of curveGet 3 of 4 questions to level up!
Derivative & the direction of a functionGet 3 of 4 questions to level up!

Estimating derivatives with secant lines and average rate of change

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Secant lines & average rate of change
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Estimating derivatives
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Practice
Secant lines & average rate of changeGet 3 of 4 questions to level up!
Estimate derivativesGet 3 of 4 questions to level up!

Quiz 1

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Secant lines & average rate of change with arbitrary points

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Secant line with arbitrary difference
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Secant line with arbitrary point
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Secant line with arbitrary difference (with simplification)
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Secant line with arbitrary point (with simplification)
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Practice
Secant lines & average rate of change with arbitrary pointsGet 3 of 4 questions to level up!
Secant lines & average rate of change with arbitrary points (with simplification)Get 3 of 4 questions to level up!

Formal definition of the derivative as a limit

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Formal definition of the derivative as a limit
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Formal and alternate form of the derivative
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Worked example: Derivative as a limit
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Worked example: Derivative from limit expression
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The derivative of x² at x=3 using the formal definition
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The derivative of x² at any point using the formal definition
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Practice
Derivative as a limitGet 3 of 4 questions to level up!

Differentiability

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Differentiability and continuity
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Differentiability at a point: graphical
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Differentiability at a point: algebraic (function is differentiable)
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Differentiability at a point: algebraic (function isn't differentiable)
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Proof: Differentiability implies continuity
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Practice
Differentiability at a point: graphicalGet 3 of 4 questions to level up!
Differentiability at a point: algebraicGet 3 of 4 questions to level up!

Tangent lines and local linearity

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The derivative & tangent line equations
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Finding tangent line equations using the formal definition of a limit
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Local linearity
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Local linearity and differentiability
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Practice
The derivative & tangent line equationsGet 3 of 4 questions to level up!
Approximation with local linearityGet 3 of 4 questions to level up!

Optional videos

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Proof: Differentiability implies continuity
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Quiz 2

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Unit test

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About this unit

The derivative of a function describes the function's instantaneous rate of change at a certain point. Another common interpretation is that the derivative gives us the slope of the line tangent to the function's graph at that point. Learn how these two interpretations describe the same thing and how we can define the derivative formally using... limits!