Advertisements
Advertisements
प्रश्न
Find `dy/dx` in the following:
`y = sin^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`
उत्तर
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
Putting x`= tan theta`,
`therefore y = sin^-1 ((1 - tan^2 theta)/(1 + tan^2 theta)) = sin^-1 (cos 2 theta)`
`y = sin^-1 sin (pi/2 - 2 theta)`
`y = pi/2 - 2theta`
`= pi/2 - 2 tan^-1 x`
On differentiating with respect to x,
`dy/dx = pi/2 d/dx (1) - 2 d/dx tan^-1 x`
`= 0 - 2 xx 1/(1 + x^2) = - 2/(1 + x^2)`
`= dy/dx = -2/(1 + x^2)`
APPEARS IN
संबंधित प्रश्न
If `sec((x+y)/(x-y))=a^2. " then " (d^2y)/dx^2=........`
(a) y
(b) x
(c) y/x
(d) 0
If `y=sin^-1(3x)+sec^-1(1/(3x)), ` find dy/dx
Differentiate `tan^(-1)(sqrt(1-x^2)/x)` with respect to `cos^(-1)(2xsqrt(1-x^2))` ,when `x!=0`
Find : ` d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`
Find the derivative of the following function f(x) w.r.t. x, at x = 1 :
`f(x)=cos^-1[sin sqrt((1+x)/2)]+x^x`
if `y = sin^(-1)[(6x-4sqrt(1-4x^2))/5]` Find `dy/dx `.
Find `dy/dx` in the following:
`y = tan^(-1) ((3x -x^3)/(1 - 3x^2)), - 1/sqrt3 < x < 1/sqrt3`
Find `dy/dx` in the following:
`y = cos^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`
Find `dx/dy` in the following:
`y = cos^(-1) ((2x)/(1+x^2)), -1 < x < 1`
Find `dy/dx` in the following:
`y = sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x < 1/sqrt2`
Find `dy/dx` in the following:
`y = sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`
Differentiate w.r.t. x the function:
`(sin x - cos x)^(sin x - cos x), pi/4 < x < (3pi)/4`
If `xsqrt(1+y) + y sqrt(1+x) = 0`, for, −1 < x <1, prove that `dy/dx = 1/(1+ x)^2`
Differentiate `tan^(-1) ((1+cosx)/(sin x))` with respect to x
if `x = tan(1/a log y)`, prove that `(1+x^2) (d^2y)/(dx^2) + (2x + a) (dy)/(dx) = 0`
Solve `cos^(-1)(sin cos^(-1)x) = pi/2`
If y = (sec-1 x )2 , x > 0, show that
`x^2 (x^2 - 1) (d^2 y)/(dx^2) + (2x^3 - x ) dy/dx -2 = 0`
If y = `(sin^-1 x)^2,` prove that `(1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.`
`lim_("h" -> 0) (1/("h"^2 sqrt(8 + "h")) - 1/(2"h"))` is equal to ____________.
`"d"/"dx" {"cosec"^-1 ((1 + "x"^2)/(2"x"))}` is equal to ____________.
If `"y = sin"^-1 ((sqrt"x" - 1)/(sqrt"x" + 1)) + "sec"^-1 ((sqrt"x" + 1)/(sqrt"x" - 1)), "x" > 0, "then" "dy"/"dx"` is ____________.
If y `= "cos"^2 ((3"x")/2) - "sin"^2 ((3"x")/2), "then" ("d"^2"y")/("dx"^2)` is ____________.
The derivative of sin x with respect to log x is ____________.
If y = sin–1x, then (1 – x2)y2 is equal to ______.
Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then ______.
Differentiate `sec^-1 (1/sqrt(1 - x^2))` w.r.t. `sin^-1 (2xsqrt(1 - x^2))`.
