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 d y d x

Solution:
y = (1 + 4x)3(3x2 – 1)4
d y d x
= (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 ) = 3 x 4 x 2 1   ,  find  f ( x ) . .
Solution:
f ( x ) = 3 x 4 x 2 1 = 3 x ( 4 x 2 1 ) 1 2 f ( x ) = 3 x . 1 2 ( 4 x 2 1 ) 1 2 .8 x + ( 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 = 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

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