Find the derivative of `sec^(-1)((1)/(2x^(2)-1))" w.r.t. "sqrt(1-x^(2))" at "x=(1)/(2).`

Find the derivative of `sec^(-1)((1)/(2x^(2)-1))" w.r.t. "sqrt(1-x^(2)

1.	Find the derivative of `sec^(-1)((1)/(2x^(2)-1))" w.r.t. "sqrt(1-x^(2))" at "x=(1)/(2).`
Answer» Correct Answer - 4 `"Let "y=sec^(-1)((1)/(2x^(2)-1))and z= sqrt(1-x^(2))` Put `x= cos theta. ` Therefore, `y=sec^(-1) (sec 2theta)=2theta and z=sqrt(1-cos^(2)theta)=sin theta` `therefore" "(dy)/(d""theta)=2, (dz)/(d""theta)=c os theta or (dy)/(dz)=(2)/(cos theta)=(2)/(x)` `"At "x=(1)/(2), (dy)/(dz)=4`

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