Math 171 Test 2 Solutions
Find the following derivatives. Simplify if possible, but don’t do anything silly.
d ⎡
2.
sin x ⎤⎦
dx ⎣
Chain Rule:
3.
1
2 sin x
× cos x =
cos x
2 sin x
d
⎡ cos ( 2 x ) ⎤⎦ Chain Rule again: − sin ( 2 x ) × 2 = −2sin ( 2 x )
dx ⎣
4.
2
2
d ⎡ − x2 ⎤
Another Chain Rule: e − x × −2 x = −2 xe − x
e
⎦
dx ⎣
5.
d
1
sin x
+ ln x cos x
[ln x sin x] Produce Rule: × sin x + ln x × cos x =
dx
x
x
6.
( x − 2) ×1 − ( x + 2) × 1 x − 2 − x − 2
d ⎡ x + 2⎤
−4
=
=
Quotient Rule:
2
2
2
⎢
⎥
dx ⎣ x − 2 ⎦
( x − 2)
( x − 2)
( x − 2)
d sin x
⎡ x ⎤⎦ Logarithmic Differentiation:
dx ⎣
d
⎛ sin x
⎞
xsin x × [sin x ln x ] = x sin x × ⎜
+ ln x cos x ⎟
dx
⎝ x
⎠
Note that this was the answer to question #5.
7.
d ⎡ sin x + 2 ⎤
Same as Question #4 with x replaced by sinx. Therefore, by the Chain
dx ⎢⎣ sin x − 2 ⎥⎦
−4
−4 cos x
Rule and the answer to #4:
× cos x =
2
2
( sin x − 2 )
( sin x − 2 )
8.
9. Find the first and second derivatives of the function f ( x) = x sin x .
Product Rule: f ′ ( x ) = 1 × sin x + x cos x = sin x + x cos x
Product Rule again: f ′′ ( x ) = cos x + 1 × cos x + x × − sin x = − x sin x + 2 cos x
10. Explain in clear English why the derivative of the function f ( x) = 10 x is not x × 10 x −1
(i.e. why the power rule does not apply here.)
Because the variable is in the exponent!
11. What is the derivative of f ( x) = 10 x ?
Logarithmic differentiation or memory:
d
f ′ ( x ) = 10 x × [ x ln10] = 10 x × ln10
dx
12. What is the derivative of log10 x ?
1
x ln10
13. Find the linear approximation to the function f ( x) =
1
1
= 1, f ′ ( 2 ) = −
= −1 ⇒ L ( x ) = 1 + − 1 ( x − 2 ) = 1 − x + 2 = 3 − x
2
2 −1
( 2 − 1)
f ( 2) =
14. Find the slope of the line tangent to the curve
1
2 x
+
1
at a = 2
x −1
1
2 y
1
y′ = 0 ⇒
y′ = −
1
⇒ y′ = −
x + y = 5 at the point (4,9).
y
.
x = 4, y = 9, y ′ = −
9
=−
2 y
2 x
x
4
d
1
d
⎡⎣ tan −1 ( x) ⎤⎦ =
and the chain rule to find ⎡⎣ tan −1 (e x ) ⎤⎦ .
15. Use the fact that
2
dx
dx
1+ x
1
( )
1 + ex
2
× ex =
ex
1 + e2 x
16. Let f ( x ) = sin x, g ( x ) = ln x, h ( x ) = x
f ′ ( x ) = cos x, g ′ =
1
1
, h′ ( x ) =
x
2 x
f D g D h ( x ) = sin ln x
17. Use the chain rule to find the derivative of f D g D h ( x )
(
)
cos ln x ×
1
x
×
1
2 x
=
(
cos ln x
2x
)
3
2
18. For f, g and h above, find the derivative of their product. That is, find
d ⎡
sin x ⋅ ln x ⋅ x ⎤⎦
dx ⎣
There are many ways to proceed. Since we have seen the derivative of sin x ln x twice
before, we can use that as one piece for the product rule. Or, use
( fgh )′ = f ′gh + fg ′h + fgh′
1
sin x x
sin x ln x
⎛ sin x
⎞
+ ln x cos x ⎟ x + ( sin x ln x )
=
+ ln x cos x x +
First way: ⎜
x
2 x
2 x
⎝ x
⎠
19. Suppose z = ln y, and
dy
dz
= 6 x . What is
? (Your answer will have an x and a y
dx
dx
in it.)
dz dz dy 1
6x
=
×
= × 6x =
dx dy dx y
y
20. Suppose f 2 ( t ) + g 2 ( t ) = 169 and g ′ ( t ) = 3. What is f ′ ( t ) when g ( t ) = 5?
− gg ′
2 ff ′ + 2 gg ′ = 0 ⇒ f ′ =
Put g = 5, g ′ = 3, f = 12 or − 12 since 122 + 52 = 169
f
(
Get f ′ = ±
15
12
)