Analysis and Approaches for IBDP Maths Ebook 2 | Page 133

40
Paper 1 Section A – First Principles
Example
(a)
(b)
Expand
( x )
4
� � .
Hence, find the derivative of
y
4
� from first principles.
x
[2]
[3]
Solution
(a)
(b)
4 4 �4� 3 �4� 2 2 � 4�
3 4
( x ��)
� x � � � x � � � � x � � � � x� ��
(M1) for valid expansion
�1� � 2� � 3�
( x ) x 4x 6x 4x
4 4 3 2 2 3 4
�� � � � � � � � � �
A1
d y ( x �h)
�x
� lim
dx
h�0
h
4 4
4 3 2 2 3 4 4
dy x � 4x h � 6x h � 4xh � h � x
� lim
(A1) for substitution
dx
h�0
h
3 2 2 3 4
dy 4x h � 6x h � 4xh � h
� lim
dx
h�0
h
dy
3 2 2 3
� lim(4x � 6x h � 4 xh � h )
A1
dx
h�0
dy
3 2
4x 6 x (0) 4 x(0) 0
dx � � � �
dy
3
4x
dx � A1 [3]
[2]
10
www.seprodstore.com
123