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Day-2-filled - Math 1314 Notes Week 2 Recall Finding the...

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Math 1314 Notes Week 2 Recall: Finding the slope of the tangent line (derivative) using the limit definition of the derivative. Review Example 1: (a) Find the derivative: 2 ( ) 3 5 7 f x x x = - + . (b) Find the equation of the line tangent to the graph of f at the point (2, 9). (c) Find all values of x for which the line tangent to f is horizontal.
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Section 11.1 We can use the limit definition of the derivative to find the derivative of every function, but it isn’t always convenient. Fortunately, there are some rules for finding derivatives which will make this easier. First, a bit of notation: [ ] ) ( x f dx d is a notation that means “the derivative of f with respect to x , evaluated at x .” Rule 1: The Derivative of a Constant [ ] , 0 = c dx d where c is a constant. Example 1 : If , 17 ) ( - = x f find ) ( ' x f .
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Rule 2: The Power Rule [ ] 1 - = n n nx x dx d for any real number n Example 2 : If , ) ( 5 x x f = find ) ( ' x f . Example 3 : If x x f = ) ( , find ) ( ' x f .
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Example 4 : If , 1 ) ( 3 x x f = find ) ( ' x f .
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Rule 3: Derivative of a Constant Multiple of a Function [ ] [ ] ) ( ) ( x f dx d c x cf dx d = where c is any real number Example 5 : If , 3 ) ( 4 x x f - = find ) ( ' x f . Rule 4: The Sum/Difference Rule [ ] [ ] [ ] ) ( ) ( ) ( ) ( x g dx d x f dx d x g x f dx d ± = ± Example 6 : Find the derivative: . 7 4 2 4 ) ( 2 3 x x x x f + - - =
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Rule 5: The Derivative of the Exponential Function [ ] x x e e dx d = Example 7 : Find the derivative: x e x x x f 6 2 4 ) ( 3 + - + = Rule 6: The Derivative of an Exponential Function (base is not e ) [ ] ( 29 x x a a a dx d = ln Example 8: Find the derivative: x x f 4 ) ( =
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Rule 7: The Derivative of the Logarithmic Function [ ] x x dx d 1 | | ln = , provided 0 x Example 9 : Find the derivative: ) ln( 6 2 5 ) ( x x x f - - = Section 11.2 Rule 8: The Product Rule [ ] ) ( ' ) ( ) ( ' ) ( ) ( ) ( x f x g x g x f x g x f dx d + =
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Example 1 : Use the product rule to find the derivative if ). 6 )( 5 7 ( ) ( 3 2 + - = x x x f
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Example 2 : Find the derivative if 2 ( ) x f x x e = .
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