Advertisements
Advertisements
प्रश्न
If `"y" = (sin^-1 "x")^2, "prove that" (1 - "x"^2) (d^2"y")/(d"x"^2) - "x" (d"y")/(d"x") - 2 = 0`.
Advertisements
उत्तर
Here,
`"y" = (sin^-1 "x")^2`
Now,
`"y"_1 = 2 sin^-1 "x" (1)/(sqrt(1 - "x"^2)`
⇒ `"y"_2 = (2)/(1 - "x"^2) + (2"x" sin^-1 "x")/((1 - "x"^2)^(3/2)`
⇒ `"y"_2 = (2)/(1 - "x"^2) + (2"x" sin^-1 "x")/((1 -"x"^2) sqrt(1 - "x"^2)`
⇒ `"y"_2 = (2)/(1 -"x"^2) + ("xy"_1)/((1 - "x"^2)`
⇒ `"y"_2 (1 - "x"^2) = 2 + "xy"_1`
⇒ `"y"_2 (1 - "x"^2) - "xy"_1 - 2 = 0`
Therefore, `(1 - "x"^2) (d^2"y")/(d"x"^2) - "x" (d"y")/(d"x") - 2 = 0`
Hence Proved.
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)))`
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) ((2x)/(1+x^2))`, −1 < x < 1
Find `bb(dy/dx)` in the following:
y = `sec^(-1) (1/(2x^2 - 1)), 0 < 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 `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
Solve `cos^(-1)(sin cos^(-1)x) = pi/2`
Find \[\frac{dy}{dx}\] at \[t = \frac{2\pi}{3}\] when x = 10 (t – sin t) and y = 12 (1 – cos t).
If y = sin-1 x + cos-1x find `(dy)/(dx)`.
`lim_("x"-> 0) ("cosec x - cot x")/"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 ______.
