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Question
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
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Solution
Let, y = (x + 3)2 · (x + 4)3 · (x + 5)4
Taking logarithm of both sides,
log y = log [(x + 3)2 · (x + 4)3 · (x + 5)4]
= log (x + 3)2 + log (x + 4)3 + log (x + 5)4 ...[∵ log mn = log m + log n]
= 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5) ...[∵ log mn = n log m]
Differentiating both sides with respect to x,
`1/y dy/dx = 2 d/dx log (x + 3) + 3 d/dx log (x + 4) + 4 d/dx log (x + 5)`
`1/y dy/dx = 2 * 1/(x + 3) d/dx (x + 3) + 3 xx 1/(x+ 4) d/dx (x + 4) + 4 xx 1/(x + 5) d/dx (x + 5)`
`1/y dy/dx = (2(1 + 0))/(x + 3) + (3(1 + 0))/("x" + 4) + (4(1 + 0))/(x + 5)`
`dy/dx = y [2/(x + 3) + 3/(x + 4) + 4/(x + 5)]`
`= y [(2 (x + 4) (x + 5) + 3 (x + 5) + 4 (x + 3) (x + 4))/((x + 3) (x + 4) (x + 5))]`
`= (x + 3)^2 (x + 4)^3 (x + 5)^4 xx [(2 (x^2 + 9x + 20) + 3(x^2 + 8x + 15) + 4 (x^2 + 7x + 12))/((x + 3) (x + 4) (x + 5))]`
`⇒ dy/dx= (x + 3) (x + 4)^2 (x + 5)^3 [9x^2 + 70x + 133]`
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