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# 14-6 - Math 200 Spring 2010 Handout 14 Section 14-6 The...

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Math 200, Spring 2010 Handout 14 Section 14-6 The Chain Rule The Chain rule (Case 1) Suppose that ( ) , z f x y = is a differentiable function of x and y , where ( ) x x t = and ( ) y y t = are both differentiable function of t . Then z is a differentiable function of t and dz f dx f dy dt x dt y dt = + . 1. Use the chain rule to find dz dt where ( ) 4 cos 4 , 5 , 1 z x y x t y t = + = = 2. The radius of a right circular cone is increasing at a rate of 1.8 in/s while its height is decreasing at a rate of 2.5 in/s. At what rate is the volume of the cone changing when the radius is 120 in. and the height is 140 in. The Chain rule (Case 2) Suppose that ( ) , z f x y = is a differentiable function of x and y , where ( ) , x x s t = and ( ) , y y s t = are differentiable function of s and t . Then z is a differentiable function of s and t and z z x z y s x s y s = + . z z x z y t x t y t = + Tree Diagram s and t are independent variables, x and y are intermediate variables, and z is the dependent variable.

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14-6 - Math 200 Spring 2010 Handout 14 Section 14-6 The...

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