Derivative of `(x+3)^(2) (x + 4)^(3) (x + 5)^(4)` with respect to x is .

Options

A. `(x + 3 )(x +4)(x + 5)^(2) (9x^(2) + 70x + 133)`
B. `(x + 3 )(x +4)^(2)(x + 5)^(3) (9x^(2) + 70x + 133)`
C. `(x + 3 )(x +4)^(2)(x + 5) (9x^(2) - 70x - 133)`
D. None of the above

Answer

B

Explanation

Let ` y = (x +3)^(2) (x +4)^(3) (x +5)^(4)`
On taking log on both sides , we get
` log y = 2 log (x-3) + (x+4) + 4 log (x+5)`
On differentiating both sides w.r.t.x, we get
`(1)/(y) (dy)/(dx) = 2 ((1)/(x +3)) (1 + 0) + 3((1)/(x +4)) (1 +0)+ 4((1)/(x +5)) (1 +0)`
`rArr (dy)/(dx) = y {((2)/(x+3))+ ((3)/(x +4)) + ((4)/(x+5))}`
` (x +3)^(2) (x +4)^(3) (x +5)^(4)`
`{(2(x + 4) (x +5) + 3(x +3)(x+5) + 4(x +3)(x +4))/((x +3)(x+4)(x +5))}`
`=(x +3)(x +4)^(2)(x +5)^(3) (9x^(2) + 70x +133)`