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Find the derivatives of the following : ...

Find the derivatives of the following :
(i) `(x^(3)3x^(2)+4)(4x^(5)+x^(2)-1)` (ii) `(9x^(2))/(x-3)`

Text Solution

Verified by Experts

Let `y=(x^(3)-3x^(2)+4)(4x^(5)+x^(2)-1)`
`(dy)/(dx)=(x^(3)-3x^(2)+4)(d)/(dx)(4x^(5)+x^(2)-1)+(4x^(5)+x^(2)-1)(d)/(dx)(x^(3)-3x^(2)+4)`
`=(x^(3)-3x^(2)+4)"["(d)/(dx) (4x^(5))+(d)/(dx)(x^(2))-(d)/(dx)(1)"]"+(4x^(5)+x^(2)-1) "[" (d)/(dx)(x^(3))-(d)/(dx)(3x^(2))+(d)/(dx)(4)"]" `
`=(x^(3)-3x^(2)+4)"["4 (d)/(dx)(x^(5))+(d)/(dx)(x^(2))-(d)/(dx)(1)"]"+(4x^(5)+x^(2)-1)"[" (d)/(dx)(x^(3))-3(d)/(dx)(x^(2))+(d)/(dx)(4)"]"`
`=(x^(3)-3x^(2)+4)"["4xx5x^(4)+2x-0"]"+(4x^(5)+x^(2)-1)"["3x^(2)-3xx2x+0"]"`
`=(x^(3)-3x ^(2)+4)(20x^(4)+2x)+(4x^(5)+x^(2)-1)(3x^(2)-6x)`
`=2x(10x^(3)+1)(x^(3)-3x^(2)+4)+3x(x-2)(4x^(5)+x^(2)-1)`
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