Chain Rule

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  1. Use the chain rule to find dwdt if w=xey/z , x=t2 , y=1t , and z=1+2t .
    ey/z2txz2xyz2
  2. Use the chain rule to find zs and zt if z=x2+xy+y2 , x=s+t , and y=st .
    zs=2x+y+xt+2yt ,
    zt=2x+y+xs+2ys .
  3. Use the chain rule to find zs and zt if z=sinαtanβ , α=3s+t , and β=st .
  4. If z=fxy where f is differentiable, x=gt , y=ht , g3=2 , g3=5 , h3=7 , h3=4 , fx27=6 , and fy27=8 , find dzdt when t=3 .
    62
  5. Let Wst=Fustvst , where F , u , and v are differentiable, u10=2 , us10=2 , ut10=6 , v10=3 , vs10=5 , vt10=4 , Fu23=1 , Fv23=10 . Find Ws10 and Wt10 .
    1. Suppose that f is a differentiable function of x and y and

      guv=feu+sinveu+cosv .

      Use the table of values to calculate gu00 and gv00 .

    2. Suppose that f is a differentiable function of x and y and

      grs=f2rss24r .

      Use the table of values to calculate gr12 and gs12 .

    f g fx fy
    00 3 6 4 8
    12 6 3 2 5

    1. 7 and 2 .
  6. Use the tree diagram to write out the chain rule for differentiating u=fxy , where x=xrst and y=yrst .
    ur=uxxr+uyyr ,
    us=uxxs+uyys ,
    ut=uxxt+uyyt .
  7. Let z=x2+xy3 , x=uv2+w3 , and y=u+vew . Use the chain rule to find zu , zv , and zw at uvw=210 .
    85 , 178 , and 54 .
  8. The temperature at a point xy is Txy , measured in kelvin. A Mars Exploration Rover crawls so that its position after t seconds is given by x=1+t and y=2+13t , where x and y are in centimeters. The temperature function satisfies Tx23=4 and Ty23=3 . How fast is the temperature rising on the rover's path after 3 seconds?
    2 [K/s].
  9. Soylent Green production in a given year, W , depends on the average temperature T and the annual rainfall R . It is estimated that the average temperature is rising at a rate of 0.15 K/year and rainfall is decreasing at a rate of 0.1 cm/year. It is also known that at current production levels, WT=2 and WR=8 . What is the significance of the signs of these partial derivatives? Estimate the current rate of change of Soylent Green production, dWdt .
  10. The radius of a right circular cone is increasing at a rate of 1.8 cm/s and its height is decreasing at the rate of 2.5 cm/s. At what rate is the volume of the cone changing when the radius is 120 cm and the height is 140 cm?
    Approximately 25635 cubic cm per second.
  11. The length l , width w , and height h of a rectangular box change with respect to time t . At a certain time the dimensions are lwh=122 . At the same time, l and w are increasing at a rate of 2 m/s, while h is decreasing at a rate of 3 m/s. Find the rates at which the following are changing:
    1. The volume of the box.
    2. The surface area of the box.
    3. The length of the diagonal of the box.
    1. 6 cubic meters per second.
    2. 10 square meters per second.
    3. 0 m/s.
  12. Assuming that f is differentiable and z=fxy , show that zx+zy=0 .