What is the Derivative of Cos u(x) ? The derivative of cos u(x) is -u'(x).sin u(x).
d x d [ cos u ( x )] = − d x d [ u ( x )] . sin u ( x )
Proof of Derivative of Cos u(x) Way 1 f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) [ cos u ( x ) ] ′ = h → 0 lim h cos u ( x + h ) − cos u ( x ) c o s p − c o s q = − 2. s i n ( 2 p + q ) . s i n ( 2 p − q ) [ cos u ( x ) ] ′ = h → 0 lim h − 2. sin [ 2 u ( x + h ) + u ( x ) ] . sin [ 2 u ( x + h ) − u ( x ) ] [ cos u ( x ) ] ′ = − h → 0 lim h 2. sin [ 2 u ( x + h ) + u ( x ) ] . sin [ 2 u ( x + h ) − u ( x ) ]
[ cos u ( x ) ] ′ = − h → 0 lim { h 2 . sin [ 2 u ( x + h ) + u ( x ) ] . sin [ 2 u ( x + h ) − u ( x ) ] }
[ cos u ( x ) ] ′ = − h → 0 lim { h 2 . sin [ 2 u ( x + h ) + u ( x ) ] . sin [ 2 u ( x + h ) − u ( x ) ] . u ( x + h ) − u ( x ) u ( x + h ) − u ( x ) } [ cos u ( x ) ] ′ = − h → 0 lim { u ( x + h ) − u ( x ) 2 . sin [ 2 u ( x + h ) − u ( x ) ] . h u ( x + h ) − u ( x ) . sin [ 2 u ( x + h ) + u ( x ) ] } [ cos u ( x ) ] ′ = − h → 0 lim { 2 u ( x + h ) − u ( x ) sin [ 2 u ( x + h ) − u ( x ) ] . h u ( x + h ) − u ( x ) . sin [ 2 u ( x + h ) + u ( x ) ]} [ cos u ( x ) ] ′ = − h → 0 lim 2 u ( x + h ) − u ( x ) sin [ 2 u ( x + h ) − u ( x ) ] . h → 0 lim h u ( x + h ) − u ( x ) . h → 0 lim sin [ 2 u ( x + h ) + u ( x ) ] h = 2 u ( x + h ) − u ( x ) ( h → 0 ) [ cos u ( x ) ] ′ = − h → 0 lim h sin h . h → 0 lim h u ( x + h ) − u ( x ) . h → 0 lim sin [ 2 u ( x + h ) + u ( x ) ]
t → 0 l i m t s i n t = 1
[ cos u ( x ) ] ′ = − 1. u ′ ( x ) . sin [ 2 u ( x + 0 ) + u ( x ) ]
[ cos u ( x ) ] ′ = − 1. u ′ ( x ) . sin [ 2 u ( x ) + u ( x ) ]
[ cos u ( x ) ] ′ = − 1. u ′ ( x ) . sin [ 2 2 . u ( x ) ]
[ cos u ( x ) ] ′ = − u ′ ( x ) . sin u ( x )
Way 2 By making use of the infinite series expansions of the sin u(x) and cos u(x) functions, we can prove that the derivative of cos u(x) is equal to -u'(x).sin u(x). The infinite series expansions of sin u(x) and cos u(x) functions are as follows.
s i n u ( x ) = u ( x ) − 3 ! [ u ( x ) ] 3 + 5 ! [ u ( x ) ] 5 − 7 ! [ u ( x ) ] 7 + 9 ! [ u ( x ) ] 9 − ...
c o s u ( x ) = 1 − 2 ! [ u ( x ) ] 2 + 4 ! [ u ( x ) ] 4 − 6 ! [ u ( x ) ] 6 + 8 ! [ u ( x ) ] 8 − ...
cos u ( x ) = 1 − 2 ! [ u ( x ) ] 2 + 4 ! [ u ( x ) ] 4 − 6 ! [ u ( x ) ] 6 + 8 ! [ u ( x ) ] 8 − ...
[ cos u ( x ) ] ′ = { 1 − 2 ! [ u ( x ) ] 2 + 4 ! [ u ( x ) ] 4 − 6 ! [ u ( x ) ] 6 + 8 ! [ u ( x ) ] 8 − ... } ′
[ cos u ( x ) ] ′ = ( 1 ) ′ − { 2 ! [ u ( x ) ] 2 } ′ + { 4 ! [ u ( x ) ] 4 } ′ − { 6 ! [ u ( x ) ] 6 } ′ + { 8 ! [ u ( x ) ] 8 } ′ − ...
[ cos u ( x ) ] ′ = 0 − 2 ! 2. u ( x ) . u ′ ( x ) + 4 ! 4. [ u ( x ) ] 3 . u ′ ( x ) − 6 ! 6. [ u ( x ) ] 5 . u ′ ( x ) + 8 ! 8. [ u ( x ) ] 7 . u ′ ( x ) − ...
[ cos u ( x ) ] ′ = − 2 .1 ! 2 . u ( x ) . u ′ ( x ) + 4 .3 ! 4 . [ u ( x ) ] 3 . u ′ ( x ) − 6 .5 ! 6 . [ u ( x ) ] 5 . u ′ ( x ) + 8 .7 ! 8 . [ u ( x ) ] 7 . u ′ ( x ) − ...
[ cos u ( x ) ] ′ = − 1 u ( x ) . u ′ ( x ) + 3 ! [ u ( x ) ] 3 . u ′ ( x ) − 5 ! [ u ( x ) ] 5 . u ′ ( x ) + 7 ! [ u ( x ) ] 7 . u ′ ( x ) − ...
[ cos u ( x ) ] ′ = − u ( x ) . u ′ ( x ) + 3 ! [ u ( x ) ] 3 . u ′ ( x ) − 5 ! [ u ( x ) ] 5 . u ′ ( x ) + 7 ! [ u ( x ) ] 7 . u ′ ( x ) − ...
[ cos u ( x ) ] ′ = − u ′ ( x ) . { u ( x ) − 3 ! [ u ( x ) ] 3 + 5 ! [ u ( x ) ] 5 − 7 ! [ u ( x ) ] 7 + ... }
[ cos u ( x ) ] ′ = − u ′ ( x ) . sin u ( x )
Q u es t i o n :
f ( x ) = c o s ( 3 x 3 − 2 x 2 ) ⇒ f ′ ( x ) = ?
A n s w er :
f ( x ) = c o s ( 3 x 3 − 2 x 2 )
f ′ ( x ) = [ c o s ( 3 x 3 − 2 x 2 ) ] ′
f ( x ) = c o s u ( x ) ⇒ f ′ ( x ) = − u ′ ( x ) . s i n u ( x )
f ′ ( x ) = − ( 3 x 3 − 2 x 2 ) ′ . s i n ( 3 x 3 − 2 x 2 )
f ′ ( x ) = − ( 3.3. x 2 − 2.2. x ) . s i n ( 3 x 3 − 2 x 2 )
f ′ ( x ) = − ( 9 x 2 − 4 x ) . s i n ( 3 x 3 − 2 x 2 )