Analysis and Approaches for IBDP Maths Ebook 2 | Page 135

41
Paper 1 Section A – Inverse Trigonometric Functions
Example
By considering the derivative of � x , show that the derivative of arcsin � x is
�
.
2 2
1��
x
[5]
Solution
Let
sin y
y� arcsin � x.
� � x
A1
d d
(sin y) ( x)
dx
� dx �
d(sin y) dy d( � x)
� � A1
dy dx dx
d
cos y
� �
d A1
dy
�
�
dx
cos y
M1
d
� (arcsin � x ) �
dx
�
1�sin
d �
(arcsin � x ) �
dx
1 � ( � x)
d �
(arcsin � x ) �
dx
1��
x
2 2
2
2
y
A1
AG
[5]
10
Exercise 41
e
1. By considering the derivative of ex , show that the derivative of arctanex is
2 2
1� ex
.
[5]
x x 1
2. By considering the derivative of , show that the derivative of arccos is �
3 3
2
9 � x
.
[5]
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