Short Questions (Question 8 – 10)


Question 8:
Find d s d t for each of the following functions.
( a )   s = ( t 3 t ) 2 ( b )   s = ( t + 1 ) ( 3 5 t ) t 2

Solution:
(a)
s = ( t 3 t ) 2 s = ( t 3 t ) ( t 3 t ) s = t 2 6 + 9 t 2 s = t 2 6 + 9 t 2 d s d t = 2 t 18 t 3 = 2 t 18 t 3

(b)
s = ( t + 1 ) ( 3 5 t ) t 2 s = 3 t 5 t 2 + 3 5 t t 2 = 5 t 2 2 t + 3 t 2 s = 5 2 t + 3 t 2 = 5 2 t 1 + 3 t 2 d s d t = 2 t 2 6 t 3 = 2 t 2 6 t 3



Question 9:
Given that  y = 1 5 x 4 x 3 , find  d y d x .

Solution:
d y d x = v d u d x u d v d x v 2 = ( x 3 ) . 20 x 3 ( 1 5 x 4 ) .1 ( x 3 ) 2 d y d x = 20 x 4 + 60 x 3 1 + 5 x 4 ( x 3 ) 2 d y d x = 15 x 4 + 60 x 3 1 ( x 3 ) 2



Question 10:
Given that  f ( x ) = ( x 2 3 ) 5 1 3 x , find  f ' ( 0 ) .

Solution:
f ( x ) = ( x 2 3 ) 5 1 3 x f ' ( x ) = v d u d x u d v d x v 2 = ( 1 3 x ) .5 ( x 2 3 ) 4 .2 x ( x 2 3 ) 5 . 3 ( 1 3 x ) 2 f ' ( x ) = 10 x ( 1 3 x ) ( x 2 3 ) 4 + 3 ( x 2 3 ) 5 ( 1 3 x ) 2 f ' ( x ) = ( x 2 3 ) 4 [ 10 x 30 x 2 + 3 ( x 2 3 ) ] ( 1 3 x ) 2 f ' ( x ) = ( x 2 3 ) 4 [ 27 x 2 + 10 x 9 ] ( 1 3 x ) 2 f ' ( 0 ) = ( 0 2 3 ) 4 [ 27 ( 0 ) 2 + 10 ( 0 ) 9 ] ( 1 3 ( 0 ) ) 2 f ' ( 0 ) = 81 × ( 9 ) 1 = 729