Section 3.4ppt - n x u y )] ( [ = Examples: ( 29 3 3 2 3 8...

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Section 3.4 The Chain Rule
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One of THE MOST POWERFUL Rules of Differentiation The chain rule allows you to take derivatives of compositions of functions that may be hard or even impossible to differentiate with previous rules only. Examples: not much fun, but doable with previous techniques Previously impossible (Unless you want to try to definition) ( 29 4 2 1 ) ( + = x x f ( 29 3 2 2 1 ) ( + = x x f
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The Chain Rule If y=f(u) is a differentiable function of u, and u=g(x) is a differentiable function of x, then y=f(g(x)) is a differentiable function of x and dy/dx = dy/du du/dx OR ) ( ' )) ( ( ' )] ( ( [ x g x g f x g f dx d =
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What does this mean? When taking the derivative of a composite function, you first take the derivative of the outside function at the inner function and then multiply by the derivative of the inner function.
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Common Types 1)One of the most common types of composite functions is What the chain rule tells us to do in this case is take the derivative with respect to the outside exponent, and leave the inner function alone. Then, we multiply by the derivative of the inner function (what was being raised to the power)
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Unformatted text preview: n x u y )] ( [ = Examples: ( 29 3 3 2 3 8 ) 7 9 ( +-= + = x y t y What about Trig functions and Exponentials? dx du e e dx d dx du u u dx d dx du u u dx d dx du u u dx d u u ) ( ] [ tan) (sec ] [sec ) (sec ] [tan ) (cos ] [sin 2 = = = = dx du u u u dx d dx du u u dx d dx du u u dx d ) cot (csc ] [csc ) (csc ] [cot ) (sin ] [cos 2-=-=-= Examples x y x x y 3 tan 2 sin sin = + = Exponentials and Logarithms Let a be a positive real number not = 1 and let u be a differentiable function of x. , 1 ] [ln = u dx du y u dx d dx du u u dx d 1 ] [ln = , 1 ] [ln = x x x dx d [ ] [ ] dx du a a a dx d a a a dx d u u x x ) )( (ln ) )( (ln = = [ ] [ ] dx du u a u dx d x a x dx d a a ) (ln 1 log ) (ln 1 log = = More Examples: x y x y x y 3 3 3 6 sec ) 2 ln( = = + = And more examples The chain rule can be used in conjunction with both product and quotient rules. You will need to decide what you should do first. ( 29 2 5 2 ) ( 2 5 2 ) ( 2 5 5 2 + + = + + = x x x g x x x g...
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Section 3.4ppt - n x u y )] ( [ = Examples: ( 29 3 3 2 3 8...

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