Math 171 Test 2 Solutions
Find the following derivatives. Simplify if possible, but don’t do anything silly.
2. sin
d
x
dx ⎡⎤
⎣⎦
Chain Rule: 1c
cos
2sin 2sin
os
x
x
x
x
×=
3.
()
cos 2
d
x
dx ⎡
⎣⎤
⎦
Chain Rule again:
(
)
(
)
sin 2 2 2sin 2
x
x−×=−
4. 2
x
de
dx
−
⎡
⎣⎤
⎦
Another Chain Rule: 22
22
xx
exxe
−
−
×− =−
5.
[
ln sin
d
]
x
x
dx Produce Rule: 1sin
sin ln cos ln cos
x
x
xx x
xx
×+× = + x
6. 2
2
dx
dx x
+
⎡
⎢
−
⎣⎦
⎤
⎥
Quotient Rule:
()
(
)
() () ()
22
21 21 22 4
22
xxxx
xx
−×−+× −−− −
==
−−
2
2
x
−
7. sin x
dx
dx ⎡
⎣⎤
⎦
Logarithmic Differentiation:
[]
sin sin sin
sinln lncos
xx
dx
x
xx x x x
dx x
⎛⎞
×=×+
⎜⎟
⎝⎠
Note that this was the answer to question #5.
8. sin 2
sin 2
dx
dx x
+
⎡
⎢
−
⎣⎦
⎤
⎥
Same as Question #4 with x replaced by sinx. Therefore, by the Chain
Rule and the answer to #4:
() ()
22
44
cos
sin 2 sin 2
cos
x
x
xx
−−
×=
−−
9. Find the first and second derivatives of the function () sin
f
xxx
=
.
Product Rule:
()
1sin cos sin cos
f
xxxxxx
′=× + = + x
Product Rule again:
()
cos 1 cos sin sin 2cos
f
xx xxxxx
′′ =+×+×−=− +x
