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)