Advertisements
Advertisements
प्रश्न
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
Advertisements
उत्तर
`sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3))` = m
∴ `(7x^3 - 5y^3)/(7x^3 + 5y^3)` = sec m =k ...(Say)
∴ 7x3 – 5y3 = 7kx3 + 5ky3
∴ (5k + 5)y3 = (7 – 7k)x2
∴ `y^3/x^3 = (7 - 7k)/(5k + 5)`
∴ `y/x = ((7 - 7k)/(5k + 5))^(1/3)` = p, where p is a constant
∴ `"d"/"dx"(y/x) = "d"/"dx"(p)`
∴ `(x"dy"/"dx" - y "d"/"dx"(x))/(x^2)` = 0
∴ `x"dy"/"dx" - y xx 1` = 0
∴ `x"dy"/"dx"` = y
∴ `"dy"/"dx" = y/x` ...(1)
∴ `(d^2y)/(dx^2) = "d"/"dx"(y/x)`
= `(x"dy"/"dx" - y "d"/"dx"(x))/(x^2)`
= `(x(y/x) - y xx 1)/(x^2)` ...[By (1)]
= `(y - y)/x^2`
= `0/x^2`
= 0
Note : `"dy"/"dx" = y/x. "where" y/x` = p.
∴ `"dy"/"dx"` = p, where p is a constant.
∴ `(d^2y)/(dx^2) = "d"/"dx"(p)` = 0.
APPEARS IN
संबंधित प्रश्न
Find dy/dx if x sin y + y sin x = 0.
Find `bb(dy/dx)` in the following:
xy + y2 = tan x + y
Find `bb(dy/dx)` in the following:
x3 + x2y + xy2 + y3 = 81
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Show that the derivative of the function f given by
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Is |sin x| differentiable? What about cos |x|?
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Differentiate e4x + 5 w.r..t.e3x
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
If x = tan-1t and y = t3 , find `(dy)/(dx)`.
If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
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((cosx)/(1 + sinx)) w.r.t. sec^-1 x.`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.
Find the nth derivative of the following : (ax + b)m
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:
y = e8x . cos (6x + 7)
Choose the correct option from the given alternatives :
If y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to
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(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
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 `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
Find `(dy)/(dx)`, if `y = sin^-1 ((2x)/(1 + x^2))`
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
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`
Find `dy/dx` if, x = e3t, y = `e^sqrtt`
Find `dy/dx` if, `x = e^(3t), y = e^sqrtt`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
