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lecture13 - tangent lines Derivatives of Exponential...

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Lecture 13: Derivatives of Polynomials and Exponentials (Sec. 3.1) Derivative of a Constant If c is a constant, then d dx ( c ) = 6 - ? ± Now consider power functions of the form f ( x ) = x n : 1) d dx ( x ) = 6 - ? ±
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2) For any real number n , d dx ( x n ) =
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ex. Find the following derivatives: 1) d dx ± ± 2 ² = 2) d dx ( x 52 ) = 3) d dx ± 1 x 4 ² = 4) d 2 dx 2 ± 1 x 4 ² =
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Di±erentiate the functions: ex. f ( x ) = ± 4 p x x ² ex. y = x 5 p x 2
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ex. Find the equation of the tangent line to f ( x ) = 3 p x at x = ± 1.
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Def. The normal line to a curve at a point P is the line through P that is perpendicular to the tangent line at P . Find the equation of the normal line to f ( x ) = 3 p x at x = ± 1.
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Constant Multiple Rule If c is a constant and f is di±erentiable then d dx ( c f ( x )) = Sum and Di±erence Rules If f and g are both di±erentiable, d dx [ f ( x ) ± g ( x )] = This can be extended to
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ex. Find f 0 ( x ) if f ( x ) = 6 x 3 ± 3 x 2 ± 12 x + 4 : At which x -values does f ( x ) have horizontal tangent lines? Write the equation of one of those horizontal
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Unformatted text preview: tangent lines. Derivatives of Exponential Functions Consider d dx f ( x ) where f ( x ) = a x : f ( x ) = lim h ! Since f (0) = lim h ! then f ( x ) = What does this say about the rate of change of any exponential? If a = 2, f (0) = lim h ! If a = 3, f (0) = lim h ! De±nition: e is the number such that lim h ! e h ± 1 h = d dx ( e x ) = 6-? ± ex. At what point is the tangent line to f ( x ) = e x parallel to y = 4 x + 1? Find f ( x ) for the following functions: ex. f ( x ) = p x + 3 p x 4 p x ex. g ( x ) = ex 2 + 2 e x + xe 2 + x e 2 ex. h ( x ) = x ln 3 ± ± ln 3 ± e e...
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