Answers and Explanations 473
Answers
701–800
758.
−
−
4
9
2
x
Rewrite the arccosine as the inverse cosine:
y
x
x
=
=
−
4
3
4
3
1
arccos
cos
Now use the chain rule along with the derivative formula for cos
−1
x:
′ =
−
− ( )
y
x
4
1 3
1 3
2
Next, simplify the term underneath the square root. Begin by getting common
denominators under the radical:
′ = −
−
y
x
4
3
1
9
9
2
Then split up the square root and simplify:
′ = −
−
= −
−
= −
−
y
x
x
x
4
3
1
9
9
4
3
1
9
9
4
3
3
9
2
2
2
= −
−
4
9
2
x
759.
sin
−1 x
First, rewrite the radical using an exponent:
y x
x
x
x
x
x
=
+ −
=
+ −
(
)
−
−
sin
sin
1
2
1
2
1 2
1
1
Use the product rule on the first term and the chain rule on the second term to
get the derivative:
′ =
+
−
+
−
( ) −
( )
=
+ −
− −
=
−
−
−
−
y
x x
x
x
x
x
x
x
x
x
1
1
1
1
2
1
2
1
1
1
2
2
1 2
1
2
2
sin
sin
sin
1 1 x
Answers
701–800
758.
−
−
4
9
2
x
Rewrite the arccosine as the inverse cosine:
y
x
x
=
=
−
4
3
4
3
1
arccos
cos
Now use the chain rule along with the derivative formula for cos
−1
x:
′ =
−
− ( )
y
x
4
1 3
1 3
2
Next, simplify the term underneath the square root. Begin by getting common
denominators under the radical:
′ = −
−
y
x
4
3
1
9
9
2
Then split up the square root and simplify:
′ = −
−
= −
−
= −
−
y
x
x
x
4
3
1
9
9
4
3
1
9
9
4
3
3
9
2
2
2
= −
−
4
9
2
x
759.
sin
−1 x
First, rewrite the radical using an exponent:
y x
x
x
x
x
x
=
+ −
=
+ −
(
)
−
−
sin
sin
1
2
1
2
1 2
1
1
Use the product rule on the first term and the chain rule on the second term to
get the derivative:
′ =
+
−
+
−
( ) −
( )
=
+ −
− −
=
−
−
−
−
y
x x
x
x
x
x
x
x
x
x
1
1
1
1
2
1
2
1
1
1
2
2
1 2
1
2
2
sin
sin
sin
1 1 x