Math 1110 Prelim 2 Answers
Please find below abbreviated answers to the questions on the second prelim. On several of the
questions, more explanation would be necessary to receive full credit.
1. (
20 points
) For each of the following functions, determine the derivative with respect to
x
.
(a) (
4 points
)
f
(
x
) =
x
1
/
3
(b) (
4 points
)
f
(
x
) = log
10
2
(c) (
4 points
)
f
(
x
) = tan(sin(4
x
))
(d) (
4 points
)
f
(
x
) =
xe

x
+
e
3
x
(e) (
4 points
)
f
(
x
) =
4
x
ln
x
Solution
(a)
f
0
(
x
) =
1
3
x

2
/
3
(b)
f
0
(
x
) = 0
(c)
f
0
(
x
) = 4 cos(4
x
) sec
2
(sin(4
x
))
(d)
f
0
(
x
) =
e

x

xe

x
+ 3
e
3
x
(e)
f
0
(
x
) =
4 ln
x

4
(ln
x
)
2
2. (
15 points
) For each of the following functions, determine the derivative with respect to
x
.
(a) (
5 points
)
f
(
x
) =
x
tan(
x
)
(b) (
5 points
)
f
(
x
) =
e
sin(
√
x
)
(c) (
5 points
)
f
(
x
) = csc(
x
) tan(
x
)
Solution:
(a)
f
0
(
x
) =
x
tan
x
(sec
2
(
x
) ln
x
+
tan
x
x
)
(b)
f
0
(
x
) =
1
2
√
x
cos(
√
x
)
e
sin(
√
x
)
(c)
f
0
(
x
) = sec(
x
) tan(
x
)
3. (
15 points
) Find the slope of the tangent line to the curve
y
2
+
xy
= sin(
xy
) + 1 at the point
(0
,
1).
Solution:
Taking the derivative of the equation we get 2
y
0
y
+
xy
0
+
y
= (
y
+
y
0
x
) cos(
xy
).
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