In this, the second section of the course, the topics include:

Concept of the derivative
• Derivative presented graphically, numerically, and analytically.
• Derivative interpreted as an instantaneous rate of change.
• Derivative defined as the limit of the difference quotient.
• Relationship between differentiability and continuity.

Derivative at a point
• Slope of a curve at a point.  Examples are emphasized, including points at which
there are vertical tangents and points at which there are no tangents.
• Tangent line to a curve at a point and local linear approximation.
• Instantaneous rate of change as the limit of average rate of change.
• Approximate rate of change from graphs and tables of values.

Derivative as a function
• Corresponding characteristics of graphs of ƒ and ƒ’.
• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.
• The Mean Value Theorem and its geometric interpretation.
• Equations involving derivatives.  Verbal descriptions are translated into equations
involving derivatives and vice versa.

Second derivatives
• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ?’’.
• Relationship between the concavity of ƒ and the sign of ƒ?’’.
• Points of inflection as places where concavity changes.

## Collection Contents

by Janet Pinto

Resources in this section listed by standard.
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### D.1: Understand the concept of derivative multiple ways

by Allen Wolmer

Understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change.
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### D.2: State, understand, and apply the definition of derivative.

by Allen Wolmer

State, understand, and apply the definition of derivative.
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### D.3: Find the derivatives of various types of functions.

by Allen Wolmer

Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.
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### D.4: Find the derivatives of sums, products, and quotients.

by Allen Wolmer

Find the derivatives of sums, products, and quotients.
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### D.5: Find the derivatives of composite functions, using the chain rule.

by Allen Wolmer

D.5: Find the derivatives of composite functions, using the chain rule.
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### D.6: Find the derivatives of implicitly-defined functions.

by Allen Wolmer

Find the derivatives of implicitly-defined functions.
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### D.7: Find the derivatives of inverse functions.

by Allen Wolmer

Find the derivatives of inverse functions.
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### D.8: Find second derivatives and derivatives of higher order.

by Allen Wolmer

Find second derivatives and derivatives of higher order.
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### D.9: Find derivatives using logarithmic differentiation.

by Allen Wolmer

Find derivatives using logarithmic differentiation.
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### D.10: Understand and apply the relationship between differentiability and continuity.

by Allen Wolmer

Understand and apply the relationship between differentiability and continuity.
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### D.11: Understand and apply the Mean Value Theorem.

by Allen Wolmer

Understand and apply the Mean Value Theorem.
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### D.12: Instantaneous rate of change as the limit of average rate of change.

by Allen Wolmer

D.12: Understand instantaneous rate of change as the limit of average rate of change.
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### D.13: Approximate rate of change from graphs and tables of values.

by Allen Wolmer

Approximate rate of change from graphs and tables of values.
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### D.14: Equations involving derivatives.

by Allen Wolmer

Equations involving derivatives.  Verbal descriptions are translated into equations involving derivatives and vice versa.
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### D.15: L’Hopital’s Rule.

by Allen Wolmer

D.15: Evaluate indeterminate limits using L’Hopital’s Rule.
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