Question

D
Evaluate
arcsin
(6)]
at x = 4.
dx

Answer 1

`(1)/(2√(5) )`

Explanation:

Let,
`\text{sin}^(-1)((x)/(6))` = y

sin(y) =
`(x)/(6)`

`(d)/(dx)\text{sin(y)}=(d)/(dx)((x)/(6))`

`(d)/(dx)\text{sin(y)}=(1)/(6)`

`(d)/(dx)\text{sin(y)}=\text{cos}(y)(dy)/(dx)` ---------(1)

`(1)/(6)=\text{cos}(y)(dy)/(dx)`

`(dy)/(dx)=\frac{1}{6\text{cos(y)}}`

cos(y) =
`\sqrt{1-\text{sin}^(2)(y) }`

=
`\sqrt{1-((x)/(6))^2}`

=
`\sqrt{1-((x^2)/(36))}`

Therefore, from equation (1),

`(dy)/(dx)=\frac{1}{6\sqrt{1-(x^2)/(36)}}`

Or
`(d)/(dx)[\text{sin}^(-1)((x)/(6))]=\frac{1}{6\sqrt{1-(x^2)/(36)}}`

At x = 4,

`(d)/(dx)[\text{sin}^(-1)((4)/(6))]=\frac{1}{6\sqrt{1-(4^2)/(36)}}`

`(d)/(dx)[\text{sin}^(-1)((2)/(3))]=\frac{1}{6\sqrt{1-(16)/(36)}}`

`=\frac{1}{6\sqrt{(36-16)/(36)}}`

`=\frac{1}{6\sqrt{(20)/(36) }}`

`=(1)/(√(20))`

`=(1)/(2√(5))`