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The derivative of y=sin^(-1)((3x+sqrt(16...

The derivative of `y=sin^(-1)((3x+sqrt(16-16x^(2)))/(5))` with respect to x at `x=(sqrt(3))/(2)`, is

A

-2

B

2

C

-4

D

does not exist

Text Solution

Verified by Experts

The correct Answer is:
A
`y=sin^(-1)((3x+4sqrt(1-x^(2)))/(5))`
`"Let "x=sintheta,thetain[(-pi)/(2),(pi)/(2)],y=sin^(-1)((3sintheta+4costheta)/(5))`
`:.y=sin^(-1)(sin(theta+alpha))"where "tanalpha=(4)/(3)`
Now, `(pi)/(4)ltalphalt(pi)/(3)," when "x=(sqrt(3))/(2),theta=(pi)/(2)," ":.(pi)/(2)lttheta+alpha,:.y=pi-(theta+alpha)`
`y=pi-tan^(-1)((4)/(3))-sin^(-1)x,(dy)/(dx)=(-1)/(sqrt(1-x^(2))),:.(dy)/(dx)underset(x=(sqrt(3))/(2))" "=-2`.
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