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Math 1304 Calculus I

Math 1304 Calculus I. 2.8 – The Derivative. Definition of Derivative. Definition: The derivative of a function f at a number a, denoted by f’(a) is given by the formula. Definition of Derivative. Definition: The derivative of a function f the function whose formula is given by.

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Math 1304 Calculus I

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  1. Math 1304 Calculus I 2.8 – The Derivative

  2. Definition of Derivative • Definition: The derivative of a function f at a number a, denoted by f’(a) is given by the formula

  3. Definition of Derivative • Definition: The derivative of a function f the function whose formula is given by

  4. Definition of Derivative • An alternate formula for derivative at a point a is

  5. Graphing Example • Given a graph of a function, graph its derivative (do y=sin(x) in class)

  6. Examples • Find the derivative of

  7. Example • f(x) = |x|

  8. Differentiable • Definition (differentiable at a point): A function is said to be differentiable at a if f’(a) exists. • Definition (differentiable on an interval): A function is said to be differentiable on an interval if f’(a) exists for all points a of the interval.

  9. Differentiable implies Continuous • Theorem: If f is differentiable at a then it is continuous at a. • Note: the reverse is not true.

  10. Various Notations • There are several different ways of denoting the derivative of a function y=f(x) The symbols D and d/dx are called differentiation operators.

  11. Second Derivatives • We may take the derivative of the derivative. This is called the second derivative. • Some notation: • Alternate notations:

  12. Third Derivatives • We may continue taking derivatives to get the third, fourth, and more derivatives • Some notation for the third: • Alternate notations:

  13. Nth Derivatives • Notations for the nth derivative:

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