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Computing Partial Derivatives Algebraically Quiz

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This quiz tests the work covered in the lecture on Computing Partial Derivatives and corresponds to Section 14.2 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There are more web quizzes at Wiley, select Section 2. This quiz was the same as the Section 1 quiz at 14/12/05.

There are some good examples of partial derivatives and some easy exercises at http://www.analyzemath.com/calculus/multivariable/partial˙derivatives.html and an explanation at http://www.ucl.ac.uk/Mathematics/geomath/level2/pdiff/pd3.html and more exercises at http://www.ucl.ac.uk/Mathematics/geomath/level2/pdiff/pd4.html.


 

Question 1

 
 
Suppose f(x,y) = 3x2 + 2xy3 + 4y2. Which one of the following statements is correct?
a) fx(x,y) = 6x+ 6y2 + 8y and fy(x,y) = 3x2 + 6xy + 8y
b) fx(x,y) = 6x +2y3 + 4y2 and fy(x,y) = 6x + 6xy2 + 8y
c) fx(x,y) = 6x + 2y3 and fy(x,y) = 6xy2 + 8y
d) fx(x,y) = 6x + 6xy2 and fy(x,y) = 2y3 + 8y

 

Not correct. Choice (a) is false.
Try again, remember to treat the y ’s as a constant when differentiating with respect to x and to treat the x ’s as a constant when differentiating with respect to y .
Not correct. Choice (b) is false.
Try again, remember to treat the y ’s as a constant when differentiating with respect to x and to treat the x ’s as a constant when differentiating with respect to y.
Your answer is correct.
Not correct. Choice (d) is false.
Try again, remember to treat the y ’s as a constant when differentiating with respect to x and to treat the x ’s as a constant when differentiating with respect to y.
 

Question 2

 
 
Suppose f (x,y) = x3exy . Which one of the statements is correct?
a) ∂f-= 3x2exy + x3yexy ∂x and ∂f-= x4exy ∂y
b) ∂f-= 3x2yexy ∂x and ∂f-= 3x3exy ∂y
c) ∂f-= 3x2exy + x4exy ∂x and ∂f-= x3yexy ∂y
d) ∂f ---= 3x2exy ∂x and ∂f 3xy -∂y = xe

 

Your answer is correct.
You have correctly used the product rule when differentiating with respect to x and ∂f-= x3xexy = x4exy ∂y
Not correct. Choice (b) is false.
Try again, you must use the product rule to differentiate with respect to x and regard 3 x as a constant when you differentiate with respect to y .
Not correct. Choice (c) is false.
Try again, you have not differentiated exy correctly with respect to either variable.
Not correct. Choice (d) is false.
Try again, you must use the product rule to differential with respect to x and regard x3 as a constant when you differentiate with respect to y.
 

Question 3

 
 
Consider z = (x2y+ 3xy3)4. Which one of the following statements is correct?
a) -∂z 2 3 3 2 ∂x = 4(x y - 3xy ) (2x+ 9y ) and ∂z- 2 3 3 2 ∂y = 4(x y- 3xy )(2x + 9y)
b) -∂z= 4(x2y - 3xy3)3(2xy+ 3y3) ∂x and ∂z-= 4(x2y- 3xy3)3(x2 + 9xy2) ∂y
c) ∂z-= 4(2xy+ 3y3)3 ∂x and ∂z-= 4(x2 + 9xy2)3 ∂y
d) ∂z-= 4(x2y- 3xy3)(2xy + 3y3)3 ∂x and ∂z-= 4(x2y- 3xy3)(x2 + 9xy2)3 ∂y

 

Not correct. Choice (a) is false.
Try again, you are not differentiating the internal function correctly.
Your answer is correct.
You have used the chain rule correctly.
Not correct. Choice (c) is false.
Try again, you have not used the chain rule correctly.
Not correct. Choice (d) is false.
Try again, you are not using the chain rule correctly.
 

Question 4

 
 
Suppose sinxy- f(x,y) = x2y3 . Which of one the following statements is correct? Reduce your answer to a fraction in lowest terms.
a) fx(x,y) = y-cos-x3y 2xy and fy(x,y) = xcos2xy3 3x y
b) fx(x,y) = - xycosxy-+32sinxy (xy) and fy(x,y) = - xycosxy+243-sinxy xy
c) 2 fx(x,y) = x-cosxy---2sin-xy (xy)3 and 2 fy(x,y) = y-cosxy---3sin-xy x2y4
d) fx(x,y) = xycosxy--2-sinxy- (xy)3 and fy(x,y) = xycosxy-- 3-sinxy x2y4

 

Not correct. Choice (a) is false.
Try again, using the quotient rule.
Not correct. Choice (b) is false.
Try again, recall d(sinx)-= cosx. dx
Not correct. Choice (c) is false.
Try again, you have not differentiated sinxy correctly.
Your answer is correct.

fx(x,y) = 2 3 3 (ycosxy)(x-y)---sinxy(2xy-) (x2y3)2
  = 2 4 3 x-y-cosxy--2xy-sinxy- x4y6
  = 3 xy-(xycosxy--2-sinxy)- x4y6
  = xycosxy---2sin-xy (xy)3
and
fy(x,y) =(xcosxy)(x2y3)- sin xy(x23y2) ----------(x2y3)2-----------
  = x3y3cosxy - 3x2y2sin xy ---------x4y6---------
  = x2y2(xycosxy - 3sin xy) ---------x4y6--------
  = xy-cosxy---3sin-xy x2y4