Advertisements
Advertisements
प्रश्न
If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.
Advertisements
उत्तर
2y = `sqrt(x + 1) + sqrt(x - 1)` ...[Given] (1)
Differentiating both sides w.r.t. x, we get
∴ `2 dy/dx = d/dx (sqrt(x + 1)) + d/dx (sqrt(x - 1))`
∴ `2 dy/dx = (1)/(2sqrt(x + 1))(1 + 0) + (1)/(2sqrt(x - 1))(1 - 0)`
∴ `2 dy/dx = (1)/(2sqrt(x + 1)) + (1)/(2sqrt(x - 1)`
∴ `2 dy/dx = (sqrt(x - 1) + sqrt(x + 1))/(2sqrt(x + 1).sqrt(x - 1)`
∴ `2 dy/dx = (cancel2y)/(cancel2sqrt(x^2 - 1)` ...[By (1)]
∴ `2sqrt(x^2 - 1) dy/dx` = y
Taking square both the sides,
∴ `4(x^2 - 1).(dy/dx)^2` = y2
Differentiating both sides w.r.t. x, we get,
`4(x^2 - 1) d/dx (dy/dx)^2 + (dy/dx)^2. d/dx [4(x^2 - 1)] = 2y dy/dx`
∴ `4(x^2 - 1).2 dy/dx.(d^2y)/(dx^2) + (dy/dx)^2 . 4(2x) = 2y(dy/dx)`
Cancelling `2 dy/dx` on both sides, we get,
`4(x^2 - 1)(d^2y)/(dx^2) + 4x dy/dx` = y
∴ `4(x^2 - 1)(d^2y)/(dx^2) + 4x dy/dx - y` = 0
∴ 4(x2 – 1)y2 + 4xy1 – y = 0.
APPEARS IN
संबंधित प्रश्न
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
x2 + xy + y2 = 100
Find `bb(dy/dx)` in the following:
x3 + x2y + xy2 + y3 = 81
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Examine the differentialibilty of the function f defined by
\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Find `dy/dx` if : x = 2 cos t + cos 2t, y = 2 sin t – sin 2t at t = `pi/(4)`
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
Find `(d^2y)/(dx^2)` of the following : x = a cos θ, y = b sin θ at θ = `π/4`.
If x2 + 6xy + y2 = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.
Find the nth derivative of the following : `(1)/(3x - 5)`
Find the nth derivative of the following : y = eax . cos (bx + c)
Choose the correct option from the given alternatives :
If y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to
Choose the correct option from the given alternatives :
If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........
Choose the correct option from the given alternatives :
If x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Choose the correct option from the given alternatives :
If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are
Differentiate the following w.r.t. x : `sin^2[cot^-1(sqrt((1 + x)/(1 - x)))]`
Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`
If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.
`(dy)/(dx)` of `2x + 3y = sin x` is:-
y = `e^(x3)`
If log(x+y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx` if , x = `e^(3t), y = e^(sqrtt)`
Find `dy/dx` if, x = `e^(3t)`, y = `e^sqrtt`
If log (x+y) = log (xy) + a then show that, `dy/dx= (-y^2)/(x^2)`
Find `dy/dx` if, `x = e^(3t), y = e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx=(-y^2)/x^2`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
