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Warm Up

Warm Up. Find the slope of the tangent line to at x=2. Answer: m= -4. Derivative of a Function. 3.1. Goal. I will be able understand the relationship between a function and its derivative as well as recognize when a function will not be differentiable. New calendar . Definition.

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Warm Up

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  1. Warm Up • Find the slope of the tangent line to at x=2. • Answer: m= -4

  2. Derivative of a Function 3.1

  3. Goal • I will be able understand the relationship between a function and its derivative as well as recognize when a function will not be differentiable. • New calendar

  4. Definition • The derivative is the formula for slope of a tangent line, instantaneous speed, or velocity of an object.

  5. Formula

  6. Alternate Formula • If asked to find the derivative at a point x=a.

  7. Notation • There are many ways to denote the derivative. They can all be found at the top of page 101. • I will also give them to you now…

  8. “the derivative of f with respect to x” Y prime “the derivative of y with respect to x” “the derivative of f with respect to x” “the derivative of f of x”

  9. Note: dx does not mean d times x ! dy does not mean d times y !

  10. does not mean ! does not mean ! Note: (except when it is convenient to think of it as division.) (except when it is convenient to think of it as division.)

  11. Example • Use the definition to find the derivative of at a=1. • Answer:

  12. Graphing f’(x) • To graph the derivative, estimate the slope at a few points, then plot those values on the new graph.

  13. Example

  14. When can you not find the derivative? • Differentiability implies continuity! If a function is not continuous, it is not differentiable. • Cusps/points, vertical asymptotes, jumps/gaps

  15. Homework

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