Differentiation Short Questions (Question 6 - 10)


Question 6:
Given that f (x) = 3x2(4x 1)7, find f’(x). 

Solution:
f (x) = 3x2(4x 1)7
f’(x) = 3x2. 7(4x 1)6. 8x + (4x 1)7. 6x
f’(x) = 168x3 (4x 1)6 + 6x (4x 1)7
f’(x) = 6x (4x 1)6 [28x2+ (4x 1)]
f’(x) = 6x (4x 1)6 (32x 1)



Question 7:
Given that y = (1 + 4x)3(3x 1)4, find dy dx

Solution:
y = (1 + 4x)3(3x2 – 1)4
dy dx
= (1 + 4x)3. 4(3x2 – 1)3.6x + (3x2 – 1)4. 3(1 + 4x)2.4
= 24x (1 + 4x)3(3x2 – 1)3 + 12 (3x2 – 1)4(1 + 4x)2
= 12 (1 + 4x)2(3x2 – 1)3 [2x (1 + 4x) + (3x2 – 1)]
= 12 (1 + 4x)2(3x2 – 1)3 [2x + 8x2 + 3x2 – 1]
= 12 (1 + 4x)2(3x2 – 1)3 [11x2 + 2x  – 1]



Question 8:
Given that f(x)=3x 4 x 2 1  , find f'(x).
Solution:
f(x)=3x 4 x 2 1 =3x ( 4 x 2 1 ) 1 2 f'(x)=3x. 1 2 ( 4 x 2 1 ) 1 2 .8x+ ( 4 x 2 1 ) 1 2 .3 f'(x)=12 x 2 ( 4 x 2 1 ) 1 2 +3 ( 4 x 2 1 ) 1 2 f'(x)=3 ( 4 x 2 1 ) 1 2 [ 4 x 2 +( 4 x 2 1 ) ] f'(x)= 3( 8 x 2 1 ) ( 4 x 2 1 )


Question 9:
Given that y= 15 x 4 x3 , find  dy dx .
Solution:
dy dx = v du dx u dv dx v 2 = ( x3 ).20 x 3 ( 15 x 4 ).1 ( x3 ) 2 dy dx = 20 x 4 +60 x 3 1+5 x 4 ( x3 ) 2 dy dx = 15 x 4 +60 x 3 1 ( x3 ) 2


Question 10:
Given that f(x)= ( x 2 3 ) 5 13x , find f'(0).
Solution:
f(x)= ( x 2 3 ) 5 13x f'(x)= v du dx u dv dx v 2 = ( 13x ).5 ( x 2 3 ) 4 .2x ( x 2 3 ) 5 .3 ( 13x ) 2 f'(x)= 10x( 13x ) ( x 2 3 ) 4 +3 ( x 2 3 ) 5 ( 13x ) 2 f'(x)= ( x 2 3 ) 4 [ 10x30 x 2 +3( x 2 3 ) ] ( 13x ) 2 f'(x)= ( x 2 3 ) 4 [ 27 x 2 +10x9 ] ( 13x ) 2 f'(0)= ( 0 2 3 ) 4 [ 27 ( 0 ) 2 +10( 0 )9 ] ( 13( 0 ) ) 2 f'(0)= 81×( 9 ) 1 =729