lecture18 - the relation and get Applying the chain rule,...

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§ 3.2 The Chain Rule and the General Power Rule Math 121 Lecture 18 ! The Chain Rule ! Marginal Cost and Time Rate of Change
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• Rule for finding the derivative of a composition of two functions. • If y is a function of u and u is a function of x , then y is a function of x . • The chain rule tell us how to find the derivative of y with respect to x • We have • The derivative of is given by equivalently, dy dx = dy du " du dx dy dx = f '( g ( x )) g '( x )
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d dx g ( x ) r ( ) = n g ( x ) r " 1 ( ) d dx g ( x ) ( )
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Use the chain rule to compute the derivative of f ( g ( x )), where and . Finally, by the chain rule, Compute using the chain rule. Since y is not given directly as a function of x , we cannot compute by differentiating y directly with respect to x . We can, however, differentiate with respect to u the relation , and get Similarly, we can differentiate with respect to x
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Unformatted text preview: the relation and get Applying the chain rule, we obtain It is usually desirable to express as a function of x alone, so we substitute 2 x 2 for u to obtain dy dx = 4 x 2 2 x 2 + 1 = 2 x 2 x 2 + 1 . ( Marginal Cost and Time Rate of Change ) The cost of manufacturing x cases of cereal is C dollars, where . Weekly production at t weeks from the present is estimated to be x = 6200 + 100 t cases. (a) Find the marginal cost, (b) Find the time rate of change of cost, (c) How fast (with respect to time) are costs rising when t = 2? (a) We differentiate C ( x ). (b) To determine , we use the Chain Rule. Now we rewrite x in terms of t using x = 6200 + 100 t . (c) With respect to time, when t = 2, costs are rising at a rate of...
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lecture18 - the relation and get Applying the chain rule,...

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