12.4 Chain Rule

# 12.4 Chain Rule - 1 2 .4  C h a in  R u l e

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Unformatted text preview: 1 2 .4  C h a in  R u l e     Suppose z = f(x,y), x = g(t), and y = h(t).  Ultimately z depends  dz on t.  We want to find  dt .    df Ex.  f(x, y) = x2y,  x = 3t + 1, y = 6t2 :           Find  dt .                              Ex.  f(x, y) = ln(xy),  x = et,  y = e‐t :             Find  df dt .                  Ex.    fx = cost,  fy = sint,  x = t2 – 4t,   y = lnt :  df Find  dt .      Suppose z = f(x, y), x = g(s, t), and y = h(s, t).  Then z ultimately  depends on s and t.            u z = x e , x = , y = v ln u : Ex.  v 2y ∂z Find ∂v   Evaluate the partial derivative at (u, v) = (1, 2).                  Ex.  Suppose z = f(x, y),  x = g(s, t), and y = h(s, t)  fx(2, 1) = 4,  fy(2, 1) = 6,    gs(‐1, 3) = 3,   hs( ‐1, 3) = ‐5,  g(‐1, 3) = 2,  and h(‐1, 3) = 1  Find fs(2, 1).                Ex.  Suppose w = f(r, s, t),  r = g(x, y),  s = h(x, y), and t = k(x, y)              dz dz and Ex.  Suppose yz = ln(x + z).    Find  dx dy .  dz F =− x, dx Fz                   Fy dz =− dy Fz   u dz 2y z = x e , x = , y = v ln u .  Do:  1.  Find  du   for  v               Evaluate it for (u, v) = (1, 2).  ⎛ ∂z ⎞ ⎛ ∂z ⎞          2.  If z = f(x, y) where x = s + t and y = s – t,  find  ⎝ ∂s ⎠ ⎝ ∂t ⎠                                     Applications  Ex. The radius of a right circular cylinder is increasing at a rate  of 2 in/min and the height is increasing at 3 in/min.  How fast  is the lateral surface area changing when the radius is 10  inches and the height is 12 inches?                Ex.  The lengths x, y, z of the edges of a rectangular box are  changing with time.  At the instant in question, V = 6 m3,   dx dy dV = = 1 m / s and = 9 m 3 / s.  x = 1m, y = 2 m,  dt dt dt How fast is z changing at that instant?        ...
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## This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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