Question

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Answer 1

OPTION A:
`$ (7)/(√(5)) $`

Explanation:

Given: f(x) =
`$ √(x^2 - 4) $` and g(x) = 3x - 2

We are to find f(g(x)).

f(g(x)) = f(3x - 2)

Substituting 3x - 2 in place of x in f(x), we get

f(g(x)) =
`$ √((3x - 2)^2 - 4)$`

`$ \implies √(9x^ 2 - 12x + 4 - 4) = √(9x^2 -12x) $`

Differentiating this we get:

`$ f(g(x))^' = (1)/(2) (9x^2 - 12x)^{(-1)/(2)}} * (18x - 12) $`

At x = 3,

`$ f(g(3))^' = (1)/(2)[9(3)^2 - 12(3)]^{(-1)/(2)}} * [18(3) - 12] $`

`$ = (1)/(2) [81 - 36]^{(1)/(2)}} * 42 $`

`$ = (21)/(√(45)) $`

`$ (21)/(3√(5)) $`

`$ (7)/(√(5))} $`

Therefore, the answer is 7/√5.

Answer 2

The answer is option a

Explanation:

Given,

f(x) =
`\sqrt{x^(2)-4 }`

g(x) = 3x - 2

By substituting g(x) in f(x) we get

f(g(x)) =
`\sqrt{(3x-2)^(2) -4}`

By differentiating with respect to we get

`\frac{\mathrm{d}\sqrt{(3x-2)^(2) -4}}{\mathrm{d} x}` =
`1/2\sqrt{(3x-2)^(2)-4}`×2×(3x-2)×3

By using chain rule we get this equation

`{\mathrm{d}f(g(x))}/{\mathrm{d} x}` =
`1/2\sqrt{(3x-2)^(2) -4}`×2×(3x-2)×3

Substituting x = 3 we get

`{\mathrm{d}f(g(x))}/{\mathrm{d} x}` = 7/
`√(5)`

Answer is option a