Advertisements
Advertisements
प्रश्न
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
Advertisements
उत्तर
x = at2 and y = 2at ...(1)
Differentiating x and y w.r.t. t, we get
`(dx)/(dt) = (d)/(dt)(at^2)`
= `a d/(dt)(t^2)`
= a × 2t
= 2at ...(2)
and
`(dy)/(dt) = d/(dt)(2at)`
= `2a d/(dt)(t)`
= 2a × 1
= 2a
∴ `(dy)/(dx) = ((dy/dt))/((dx/dt)`
= `(2a)/(2at) = (1)/t`
and
`(d^2y)/(dx^2) = d/(dx)(1/t)`
= `d/(dt)(t^-1).(dt)/(dx)`
= `(-1)t^-2.(1)/((dx/dt)`
= `(-1)/t^2 xx (1)/(2at)` ...[By (2)]
= `(-1)/(2at^3)`
∴ `(d^2y)/(dx^2) = (-1)/(2at^3)`
= `(-1)/(yt^2) = (-1)/(y xx x/a)` ...[Using (1)]
∴ `xy(d^2y)/(dx^2)` = – a
∴ `xy(d^2y)/(dx^2) + a` = 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:
2x + 3y = sin y
Find `bb(dy/dx)` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Write the derivative of f (x) = |x|3 at x = 0.
Find `(dy)/(dx) if y = cos^-1 (√x)`
Discuss extreme values of the function f(x) = x.logx
If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`
Find `"dy"/"dx"`, if : x = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.
Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`
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))`.
If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
Find the nth derivative of the following:
`(1)/x`
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : sin (ax + b)
Find the nth derivative of the following : cos (3 – 2x)
Find the nth derivative of the following : `(1)/(3x - 5)`
Choose the correct option from the given alternatives :
If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?
Choose the correct option from the given alternatives :
If y = `tan^-1(x/(1 + sqrt(1 - x^2))) + sin[2tan^-1(sqrt((1 - x)/(1 + x)))] "then" "dy"/"dx"` = ...........
Choose the correct option from the given alternatives :
If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........
If y `tan^-1(sqrt((a - x)/(a + x)))`, where – a < x < a, then `"dy"/"dx"` = .........
Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.
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, sqrt"x" + sqrt"y" = sqrt"a"`
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.
State whether the following statement is True or False:
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x) = 1/(2sqrt(x)) + 1/(2sqrt(y)) = 1/(2sqrt("a"))`
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
Let y = y(x) be a function of x satisfying `ysqrt(1 - x^2) = k - xsqrt(1 - y^2)` where k is a constant and `y(1/2) = -1/4`. Then `(dy)/(dx)` at x = `1/2`, is equal to ______.
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
`"If" log(x+y) = log(xy)+a "then show that", dy/dx=(-y^2)/x^2`
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`
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`
