Chain Rule - r = g x y s = h x y t = k x y Find ∂w ∂x...

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12.4 The Chain Rule Suppose z = f ( x, y ), x = g ( t ), and y = h ( t ). Ultimately, z depends on t . We want to find dz dt . Example f ( x, y ) = x 2 y where x = 3 t - 1 and y = 6 t 2 . Find df dt . The Chain Rule Example f ( x, y ) = ln( xy ) x = e t , y = e - t . Using substitution: 1
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Suppose z = f ( x, y ), x = g ( s, t ), y = h ( s, t ). z ultimately depends on s and t : ∂z ∂s = ∂z ∂t = Example 1 z = x 2 e y , x = u v , y = v ln u . Find ∂z ∂v at ( u, v ) = (1 , 2). Example 2 Suppose f(x,y), x = g(s,t), y = h(s,t) and we know the following values: f x (2 , 1) = 4 f y (2 , 1) = 6 g s ( - 1 , 3) = 3 h s ( - 1 , 3) = - 5 g ( - 1 , 3) = 2 h ( - 1 , 3) = 1 Find f s when ( u, v ) = ( - 1 , 3). 2
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Example 3 Suppose w = f (
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Unformatted text preview: ), r = g ( x, y ), s = h ( x, y ), t = k ( x, y ). Find ∂w ∂x . Example 4 Suppose f x = cos t , f y = sin t , x = t 2-4 t , and y = ln t . Find df dt . Application The radius of a right circular cylinder is increasing at a rate of 2”/min and the height is increasing 3”/min. How fast is the lateral surface area changing when the radius is 10” and the height is 12”? 3...
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