If `y=sin^(-1){(5x+12 sqrt(1-x^(2)))/(13)},` find `(dy)/(dx).`

Explanation

We have
`y=sin^(-1){(5)/(13).x+(12)/(13).sqrt(1-x^(2))}.`
Let `(5)/(13)=sin alpha and x = cos theta.` Then,
`cos alpha=sqrt(1-(25)/(169))=sqrt((144)/(169))=(12)/(13)`
`and sqrt(1-x^(2))=sqrt(1-cos^(2)theta)=sqrt(sin^(2)theta)=sin theta.`
`therefore y=sin^(-1){sin alpha cos theta+cos alpha sin theta}`
`=sin^(-1){sin(alpha+theta)}`
`=alpha+theta = sin^(-1)/(5)/(13)+cos^(-1)x.`
`therefore(dy)/(dx)=(d)/(dx){sin^(-1).(5)/(13)+cos^(-1)x}=(d)/(dx){sin^(-1).(5)/(13)}+(d)/(dx)(cos^(-1)x)`
`={0-(1)/(sqrt(1-x^(2)))}=(-1)/(sqrt(1-x^(2))).`
Hence, `(dy)/(dx)=(-1)/(sqrt(1-x^(2))).`