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Math 180

Math 180. 3.2 – The Derivative as a Function. By modifying the definition slightly, we can consider the derivative as a function of : is _____________ at if exists. ______________ is the process of calculating the derivative .

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Math 180

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  1. Math 180 3.2 – The Derivative as a Function

  2. By modifying the definition slightly, we can consider the derivative as a function of : is _____________at if exists. ______________is the process of calculating the derivative.

  3. By modifying the definition slightly, we can consider the derivative as a function of : is _____________at if exists. ______________is the process of calculating the derivative. differentiable

  4. By modifying the definition slightly, we can consider the derivative as a function of : is _____________at if exists. ______________is the process of calculating the derivative. differentiable Differentiation

  5. Ex 1.Using the definition, differentiate .

  6. Ex 2.Using the definition, find the derivative of for . Now find the tangent line to the curve at .

  7. Ex 2.Using the definition, find the derivative of for . Now find the tangent line to the curve at .

  8. Note: There are many ways people write the derivative of : And here’s what it looks like to plug a value into the derivative:

  9. Note: There are many ways people write the derivative of : And here’s what it looks like to plug a value into the derivative:

  10. Graphing Derivatives Remember that the derivative is the slope of the tangent line.

  11. Note that only exists if both of the following limits exist: Left-hand derivative at Right-hand derivative at

  12. Note that only exists if both of the following limits exist: Left-hand derivative at Right-hand derivative at

  13. Note that only exists if both of the following limits exist: Left-hand derivative at Right-hand derivative at

  14. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  15. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  16. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  17. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  18. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  19. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  20. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  21. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  22. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  23. Theorem: If has a derivative at , then is continuous at . (That is, differentiable functions are continuous. And if a function is not continuous at a point, then it is not differentiable there.)

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