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Question
Choose the correct option from the given alternatives :
Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is
Options
`-(29)/(15)`
`(7)/(3)`
`(31)/(15)`
`(29)/(15)`
Solution
`(29)/(15)`
[Hint : Let y = `sqrt([f(x)]^2 + [g(x)]^2`
Then `"dy"/"dx" = (1)/(2sqrt([f(x)]^2 + [g(x)]^2)).[2f(x).f'(x) + 2g(x).g'(x)]`
∴ `(dy/dx)_("at" x = 1) = (1)/(2sqrt([f(1)]^2 + [g(1)]^2)).[2f(1).f'(1) + 2g(1).g'(1)]`
= `(1)/(2sqrt(9 + 16)) xx [2(3)(-1/3) + 2(-4)(-8/3)]`
= `(1)/(2sqrt(9 + 16)) xx [(- 2) + 64/3]`
= `1/10 [-2 + 64/3]`
= `1/10 [58/3]`
= `58/30`
= `29/15`
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