� Relationships between graph properties and the derivatives : 1 . f�( x) � 0 for a �x � b : f( x ) is increasing in the interval 2 . f�( x) � 0 for a �x � b : f( x ) is decreasing in the interval 3 . f�( a) � 0: ( a, f ( a )) is a stationary point of f( x ) 4 . f�( a) � 0 and f� ( x) changes from positive to negative at x� a: ( a, f ( a )) is a maximum point of f( x )
5 . f�( a) � 0 and f� ( x) changes from negative to positive at x� a: ( a, f ( a )) is a minimum point of f( x )
6 . f��( a) � 0 and f�� ( x) changes sign at x� a: ( a, f ( a )) is a point of inflexion of f( x )
� Slopes of tangents and normals : 1 . f� ( a)
: Slope of tangent at x� a
2 .
�1 f�( a)
: Slope of normal at x� a
� Differentiation by first principle : f ( x �h) � f ( x) f�( x) � lim h �0 h
� More differentiation rules :
2 f ( x) � tan x � f ( x) � sec x �
2 f ( x) � cot x � f ( x) � � cosec x f ( x) � sec x � f �( x) � sec x tan x f ( x) � cosec x � f �( x) � � cosecx cot x
( ) x x f x � a � f �( x) � a ln a
1 f ( x) � arcsin x � f �( x)
� 1� x
1 f ( x) � arctan x � f �( x)
�
2
1 � x
� Implicit differentiation : F d d
( x , y ) � G ( x , y ) � ( , ) ( , ) d F x y � x dx
G x y
2
1 f ( x) � log a x � f �( x) � xln a 1 f ( x) � arccos x � f �( x) � �
1� x
�
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