Advertisements
Advertisements
Question
Find `bb(dy/dx)` in the following:
y = `sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`
Advertisements
Solution
y = `sec^-1 (1/(2x^2 - 1))`
Let, x = cos θ
⇒ θ = cos−1 x
∴ y = `sec^-1 (1/(2 cos^2 theta - 1))`
= `sec^-1 (1/(cos 2 theta))`
= sec−1 (sec 2 θ)
= 2 θ
= 2 cos−1 x
On differentiating with respect to x,
`dy/dx = 2 d/dx cos^-1 x`
`dy/dx = 2 xx -1/(sqrt(1 - x^2))`
`dy/dx = -2/(sqrt(1 - x^2))`
APPEARS IN
RELATED QUESTIONS
Differentiate `cos^-1((3cosx-2sinx)/sqrt13)` w. r. t. x.
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
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=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x2≤1, then find dy/dx.
Find `bb(dy/dx)` in the following:
y = `cos^(-1) ((1-x^2)/(1+x^2))`, 0 < x < 1
Find `bb(dy/dx)` in the following:
y = `cos^(-1) ((2x)/(1+x^2))`, −1 < x < 1
Find `bb(dy/dx)` in the following:
y = `sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x < 1/sqrt2`
Differentiate the function with respect to x:
`cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))], 0 < x < pi/2`
Differentiate the function with respect to x:
`(sin x - cos x)^((sin x - cos x)), pi/4 < x < (3pi)/4`
Find `dy/dx`, if y = `sin^-1 x + sin^-1 sqrt (1 - x^2)`, 0 < x < 1.
If `sqrt(1-x^2) + sqrt(1- y^2)` = a(x − y), show that dy/dx = `sqrt((1-y^2)/(1-x^2))`
Find the approximate value of tan−1 (1.001).
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 = cos (sin x), show that: `("d"^2"y")/("dx"^2) + "tan x" "dy"/"dx" + "y" "cos"^2"x" = 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 ____________.
`lim_("x" -> -3) sqrt("x"^2 + 7 - 4)/("x" + 3)` is equal to ____________.
`"d"/"dx" {"cosec"^-1 ((1 + "x"^2)/(2"x"))}` is equal to ____________.
The derivative of sin x with respect to log x is ____________.
The derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `cos^-1 [(1 - x^2)/(1 + x^2)]` 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))`.
