HW 6 3 Chain Rule Implicit Differentiation Find dz dt a) as a general formula b) at the given t value 1 z x3y2 x cos t y sin t t 3 2 z arctan(xy) x t2 y ln t t 1 3 z xey x x 1 t y 3 2t t 2 Find dy dx 4 y arctan x x4 2 sin y 0 5 xy 1 x2 y2 e x2 Find the requested partial derivatives 6 x2y4 5x2z3 4xy y2z2 find z y and x y 7 exz yz 0 find z y and y x 8 2xy 4yz 3xz 0 find z x two ways (a) method 1, solve for z (b) method 2, use implicit derivatives (c) show they are equal by substituting for z Applications 9 A manufacturing company's yearly production is worth P 1 47L0 65K0 35, where L is the amount of labor (in thousands of hours) and K is the amount of capital (in million $) and P is measured in millions of $ When L 30 and K 8, the labor is decreasing at 2 thousand hours year and K is increasing at 0 5 million $ year Find dP dt (in million $ year) HW 6 3 Chain Rule Implicit Differentiation Find dz dt a) as a general formula b) at the given t value 1 z x'y x cost y sint t 1 3 2 z arctan(xy) x t' y Int t 1 3 z xev x x 1 t y 3 2t 1 2 Find dy dx 4 y arctan x x 2 sin y 0 5 xy 1 x v Find the requested partial derivatives 6 xy 5x z' 4xy y z2 find 2 and a x ay ay 7 exz yz 0 find and dy av a x 8 2xy 4yz 3xz 0 find two ways (a) method 1, solve for z (b) method 2, use implicit derivatives (c) show they are equal by substituting for Applications 9 A manufacturing company's yearly production is worth P 1 47LK03s, where L is the amount of labor (in thousands of hours) and K is the amount of capital (in million $) and P is measured in millions of $ When L 30 and K 8, the labor is decreasing at 2 thousand hours year and K is increasing at 0 5 million $ year Find de (in million $ year)

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HW 6.3 - Chain Rule & Implicit Differentiation Find dz dt a) as a general formula b) at the given t-value 1. z = x3y2

HW 6.3 - Chain Rule & Implicit Differentiation

Find dz

a) as a general formula b) at the given t-value

1. z = x3y2 ... x = cos t & y = sin t ... t = ?/3

2. z = arctan(xy) ... x=t2 & y=ln t ... t = 1

3. z = xey/x ... x=1-t & y=3-2t ... t=2

Find dy

4. y arctan x + x4.2 - sin y = 0

5. xy

1+ x2+ y2=e x2

Find the requested partial derivatives

6. x2y4 + 5x2z3 = 4xy + y2z2 ... find ? z

? y

and ? x

? y

7. exz - yz = 0 ... find ? z

? y

and ? y

? x

8. 2xy-4yz+3xz=0 ... find ? z

? x

two ways: (a) method 1, solve for z (b) method 2, use implicit

derivatives (c) show they are equal by substituting for z

Applications

9. A manufacturing company's yearly production is worth P = 1.47L0.65K0.35, where L is the amount

of labor (in thousands of hours) and K is the amount of capital (in million $) and P is measured

in millions of $. When L=30 and K=8, the labor is decreasing at 2 thousand hours/year and K is

increasing at 0.5 million $/year. Find dP

(in million $/year).

HW 6.3 - Chain Rule & Implicit Differentiation Find dz dt a) as a general formula b) at the given t-value 1. z=x'y ... x = cost & y = sint ... t= 1/3 2. z = arctan(xy) ... x=t' & y=Int ... t= 1 3. z = xev/x ... x=1-t & y=3-2t ... 1=2 Find dy dx 4. y arctan x + x 2 - sin y = 0 5. xy 1+x+v Find the requested partial derivatives 6. xy+ + 5x z' = 4xy + y z2 ... find 2 and a x ay ay 7. exz - yz =0 ... find and dy av a x 8. 2xy-4yz+3xz=0 ... find two ways: (a) method 1, solve for z (b) method 2, use implicit derivatives (c) show they are equal by substituting for = Applications 9. A manufacturing company's yearly production is worth P = 1.47LK03s, where L is the amount of labor (in thousands of hours) and K is the amount of capital (in million $) and P is measured in millions of $. When L=30 and K=8, the labor is decreasing at 2 thousand hours/year and K is increasing at 0.5 million $/year. Find de (in million $/year)