Answers and Explanations 265
Answers
401–500
401.
6
1 5 6
5
−
− (
)
 
 
x
x
ln
To find the derivative of f x
x
x
( )
(
)
= 6
5
−
, apply the product rule while applying the
chain rule to the second factor:
=
( ) +
−
(
)
=
−
[
]
−
−
−
x
x
x
x
x
( )
(ln )( )
( ln )
1 6
6
6 5
6
1 5 6
5
5
5
( )
x
f '
402.
y = πx + 3
You can begin by finding the y value, because it isn’t given:
f 0 3
0
0 3
( ) =
( )+ ( ) =
cos
π
Next, find the derivative of the function:
=
x
sin
−
+
3
π
( )
x
f '
Substitute in the given x value to find the slope of the tangent line:
+
( )
sin( )
0
3
0
= −
=
π π
f '
Now use the point-slope formula for a line to get the tangent line at x = 0:
y
y
x
− =
−
=
+
3
3
π
π
(x 0)
403.
y = x + 1
Begin by finding the derivative of the function f
 
(x) = x
2
– x + 2:
x
=
−
2 1
( )
x
f '
Substitute in the given x value to find the slope of the tangent line:
=
− =
2 1 1 1
( )
f '
Now use the point-slope formula for a line to get the tangent line at (1, 2):
y
x
y x
− =
−
= +
2 1
1
1
(
)