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lecture16

# lecture16 - f x = e 2 x cos x at x = 0 Note the following...

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Lecture 16: The Chain Rule (Sec. 3.4) Consider the functions f ( x ) = tan ± ±x 2 ² and h ( x ) = p x 2 + 4 x ± 5. How to di±erentiate? NOTE: f and h are composite functions.

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Theorem (The Chain Rule) : If g is di±erentiable at x and f is di±erentiable at g ( x ), then the composite function F = f ± g = f ( g ( x )) is di±erentiable and F 0 ( x ) = Alternatively, if y = f ( u ) and u = g ( x ) are di±eren- tiable functions, then
ex. Find d dx (1 + tan x ) 10 ex. If h ( x ) = p x 2 + 4 x ± 5, ±nd h 0 ( x ).

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These are examples of the following: Power Rule combined with the Chain Rule If n is any real number and u = g ( x ) is di±erentiable, then or ex. If g ( x ) = 4 4 p (3 ± 2 x 2 ) 3 ²nd g 0 ( x ).
ex. If f ( x ) = (2 x +3) 4 (2 ± x ) 3 , ±nd f 0 ( x ). At which x -values does the graph of f ( x ) have a horizontal tangent line?

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ex. Find and simplify f 0 ( x ) if f ( x ) = x 2 p 3 ± 2 x . Write the equation of the tangent line to f ( x ) at x = 1.
Derivatives involving Exponential Functions ex. Find: d dx ( e ± 2 x 2 +1 ) ex. Find the equation of the tangent line to

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Unformatted text preview: f ( x ) = e 2 x cos x at x = 0. Note the following: If f ( x ) = e ax for constant a , f ( x ) = ex. d dx ± (1 + e x ) 2 e x ² ex. d dx r 1 + e 2 x 1 ± e 2 x We have the following important result: For base a > 0, d dx a x = ex. Find the slope of the tangent line to f ( x ) = 4 6 x ± cos x at x = ± . ex. Find g ( x ) for g ( x ) = 3 4 x . Derivatives involving trigonometric functions ex. Find f ( x ) if f ( x ) = tan ± ±x 2 ² : ex. Find the equation of the tangent line to y = sin 3 x cos 3 x at x = ± 2 . ex. Find the derivative of f ( x ) = sec 2 (sin 4 x ). Additional Example ex. If g ( x ) = p x 2 + 3 p 2 x + 1, ±nd g (4)....
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