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Unformatted text preview: (8/17/08) Math 20C. Lecture Examples. Section 14.6. The Chain Rule If the outside function f ( x, y ) in a composite function z = f ( x ( t ) , y ( t )) is given by a specific formula constructed from the basic functions of calculus, then the derivative d dx [ f ( x ( t ) , y ( t ))] can be found with the Chain Rule for one variables. This principle is illustrated in the next example. Example 1 Find the t-derivative of z = f ( x ( t ) , y ( t )) at t = 0 where f ( x, y ) = x 2 + y 2 , x (0) = 5 , y (0) =- 1 , x prime (0) = 3, and y prime (0) =- 4. Answer: d dt [ f ( x, y )] vextendsingle vextendsingle t =0 = 19 The general Chain Rule with two variables The following Chain Rule is needed to find derivatives of composite functions in the form z = f ( x ( t ) , y ( t )) in cases where the outer function f has only a letter name. Theorem 1 The t-derivative of the composite function z = f ( x ( t ) , y ( t )) is d dt [ f ( x ( t ) , y ( t ))] = f x ( x ( t ) , y ( t )) x prime ( t...
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