Math 1210
Quiz 4
February 7th, 2014
Answer the following two (2) questions. The value of every question is
indicated at the beginning of it. You may use scratch paper, but you can
only turn in this sheet. Please write your answer in the space provided. You
have 20 minutes.
Name:
UID:
1. (10 points) Use the chain rule to compute the derivatives of these functions:
2 x
(i) (7 points) f (x) = cos 1−x
Solution:
0
f (x) = − sin
4
x2
1−x
x2
1−x
·
2x(1 − x) + x2
(1 − x)2
(Note: this is just the 4th power of the function in
(ii) (3 points) f (x) = cos
part (i), so most of the work is already done!)
Solution:
2 0
x2
x
f (x) = 4 cos
· cos
1−x
1−x
2 2 x
x
2x(1 − x) + x2
= −4 cos3
sin
·
1−x
1−x
(1 − x)2
0
3
2. (10 points) The equation
y 3 + 7y = x3
determines y as an implicit function y = f (x). Find y 0 = Dx (y) and y 00 = Dx2 (y) by
implicit differentiation.
(i) (7 points) y 0 =
Solution: Thinking of y = f (x) and differentiating the equation y 3 + 7y = x3 with
respect to x yields
3y 2 y 0 + 7y 0 = 3x2
so solving for y 0 we obtain
y0 =
3x2
3y 2 + 7
(ii) (3 points) y 00 =
Solution: We differentiate the equation 3y 2 y 0 +7y 0 = 3x2 with respect to x, bearing
in mind that y is a function of x. We obtain:
6yy 0 y 0 + 3y 2 y 00 + 7y 00 = 6x
and solving for y 00 we conclude
y 00 =
6x − 6y(y 0 )2
3y 2 + 7
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