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MATH1111 Quizzes

Computing Partial Derivatives Algebraically Quiz
Web resources available Questions

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.

Suppose f(x,y) = 3x2 + 2xy3 + 4y2. Which one of the following statements is correct? Exactly one option must be 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

Choice (a) is incorrect
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.
Choice (b) is incorrect
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.
Choice (c) is correct!
Choice (d) is incorrect
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.
Suppose f(x,y) = x3exy. Which one of the statements is correct? Exactly one option must be correct)
a)
f x = 3x2exy + x3yexy and f y = x4exy
b)
f x = 3x2yexy and f y = 3x3exy
c)
f x = 3x2exy + x4exy and f y = x3yexy
d)
f x = 3x2exy and f y = x3exy

Choice (a) is correct!
You have correctly used the product rule when differentiating with respect to x and f y = x3xexy = x4exy
Choice (b) is incorrect
Try again, you must use the product rule to differentiate with respect to x and regard x3 as a constant when you differentiate with respect to y.
Choice (c) is incorrect
Try again, you have not differentiated exy correctly with respect to either variable.
Choice (d) is incorrect
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.
Consider z = (x2y + 3xy3)4. Which one of the following statements is correct? Exactly one option must be correct)
a)
z x = 4(x2y 3xy3)3(2x + 9y2) and z y = 4(x2y 3xy3)3(2x + 9y2)
b)
z x = 4(x2y 3xy3)3(2xy + 3y3) and z y = 4(x2y 3xy3)3(x2 + 9xy2)
c)
z x = 4(2xy + 3y3)3 and z y = 4(x2 + 9xy2)3
d)
z x = 4(x2y 3xy3)(2xy + 3y3)3 and z y = 4(x2y 3xy3)(x2 + 9xy2)3

Choice (a) is incorrect
Try again, you are not differentiating the internal function correctly.
Choice (b) is correct!
You have used the chain rule correctly.
Choice (c) is incorrect
Try again, you have not used the chain rule correctly.
Choice (d) is incorrect
Try again, you are not using the chain rule correctly.
Suppose f(x,y) = sinxy x2y3 . Which of one the following statements is correct? Reduce your answer to a fraction in lowest terms. Exactly one option must be correct)
a)
fx(x,y) = ycosxy 2xy3 and fy(x,y) = xcosxy 3x2y3
b)
fx(x,y) = xycosxy + 2sinxy (xy)3 and fy(x,y) = xycosxy + 3sinxy x2y4
c)
fx(x,y) = x2 cosxy 2sinxy (xy)3 and fy(x,y) = y2 cosxy 3sinxy x2y4
d)
fx(x,y) = xycosxy 2sinxy (xy)3 and fy(x,y) = xycosxy 3sinxy x2y4

Choice (a) is incorrect
Try again, using the quotient rule.
Choice (b) is incorrect
Try again, recall d(sinx) dx = cosx.
Choice (c) is incorrect
Try again, you have not differentiated sinxy correctly.
Choice (d) is correct!
fx(x,y)=(ycosxy)(x2y3) sinxy(2xy3) (x2y3)2
 =x2y4 cosxy 2xy3 sinxy x4y6
 =xy3(xycosxy 2sinxy) x4y6
 =xycosxy 2sinxy (xy)3
and
fy(x,y)=(xcosxy)(x2y3) sinxy(x23y2) (x2y3)2
 =x3y3 cosxy 3x2y2 sinxy x4y6
 =x2y2(xycosxy 3sinxy) x4y6
 =xycosxy 3sinxy x2y4