Advertisements
Advertisements
प्रश्न
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.
उत्तर
`y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))`
Putting x2=cos2θ, we have
`y=tan^(−1) ((sqrt(1+cos2θ)+sqrt(1−cos2θ))/(sqrt(1+cos2θ)−sqrt(1−cos2θ)))`
`y=tan^(−1) ((sqrt(2cos^2theta)+sqrt(2sin^2θ))/(sqrt(2cos^2θ)−sqrt(2sin^2θ)))`
`y=tan^(-1)((costheta+sintheta)/(costheta-sintheta))y`
`=tan^(-1)((1+tantheta)/(1-tantheta))` (Dividing the numerator and denominator by cosθ)
`y=tan^(-1)((tan(pi/4)+tantheta)/(1-tan(pi/4)tantheta))`
`⇒y=tan^(−1)[tan(π/4+θ)]`
`⇒y=π/4+θ`
`∴ y=π/4+1/2cos^(−1)x^2 (x^2=cos2θ)`
Differentiating both sides with respect to x, we get
`dy/dx=0+1/2×(−1/sqrt(1−(x^2)^2))xx2x`
`⇒dy/dx=−x/sqrt(1−x^4)`
APPEARS IN
संबंधित प्रश्न
Differentiate `cos^-1((3cosx-2sinx)/sqrt13)` w. r. t. x.
If `y=sin^-1(3x)+sec^-1(1/(3x)), ` find dy/dx
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 = cos^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`
Find `dy/dx` in the following:
`y = sin^(-1) ((1-x^2)/(1+x^2)), 0 < 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:
`cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))]`, ` 0 < x < pi/2`
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`
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).
Differentiate `tan^(-1) ((1+cosx)/(sin x))` with respect to x
Find \[\frac{dy}{dx}\] at \[t = \frac{2\pi}{3}\] when x = 10 (t – sin t) and y = 12 (1 – cos t).
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 + cos-1x find `(dy)/(dx)`.
If `log (x^2 + y^2) = 2 tan^-1 (y/x)`, show that `(dy)/(dx) = (x + y)/(x - y)`
Trigonometric and inverse-trigonometric functions are differentiable in their respective domain.
`lim_("x"-> 0) ("cosec x - cot x")/"x"` is equal to ____________.
If y `= "cos"^2 ((3"x")/2) - "sin"^2 ((3"x")/2), "then" ("d"^2"y")/("dx"^2)` is ____________.
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))`.
