If `y=(sqrt(a+x)-sqrt(a-x))/(sqrt(a+x)+sqrt(a-x)),t h e n(dy)/(dx)i se q u a lto` `(a y)/(xsqrt(a^2-x^2))` (b) `(a y)/(sqrt(a^2-x^2))` `(a y)/(xsqrt(a
If
`y=(sqrt(a+x)-sqrt(a-x))/(sqrt(a+x)+sqrt(a-x)),t h e n(dy)/(dx)i se q u a lto`
`(a y)/(xsqrt(a^2-x^2))`
(b) `(a y)/(sqrt(a^2-x^2))`
`(a y)/(xsqrt(a^2-x^2))`
(d) none of these
A. `(ay)/(xsqrt(a^(2)-x^(2)))`
B. `(ay)/(sqrt(a^(2)-x^(2)))`
C. `(ay)/(xsqrt(x^(2)-a^(2)))`
D. none of these
1 Answers
`y=(sqrt(a+x)-sqrt(a-x))/(sqrt(a+x)+sqrt(a-x))`
`=((sqrt(a+x)-sqrt(a-x))^(2))/((a+x)-(a-x))`
`=((a+x)+(a-x)-2(sqrt(a^(2)-x^(2))))/(2x)`
`=(2a-2sqrt(a^(2)-x^(2)))/(2x)=(a-sqrt(a^(2)-x^(2)))/(x)" (1)"`
Differentiating w.r.t. x, we get
`(dy)/(dx)=(x[-(1)/(2sqrt(a^(2)-x^(2)))(-2x)]-(a-sqrt(a^(2)-x^(2))))/(x^(2))`
`=(x^(2)-asqrt(a^(2)-x^(2)+a^(2)-x^(2)))/(x^(2)sqrt(a^(2)-x^(2)))=(a(a-sqrt(a^(2)-x^(2))))/(x^(2)sqrt(a^(2)-x^(2)))`
`(a)/(xsqrt(a^(2)-x^(2)))[(a-sqrt(a^(2)-x^(2)))/(x)]=(ay)/(xsqrt(a^(2)-x^(2)))" [by (1)]"`