Advertisements
Advertisements
प्रश्न
Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)
Advertisements
उत्तर
x = a(θ – sin θ), y = a(1 – cos θ)
Differentiating x and y w.r.t. θ, we get
`"dx"/"dθ" = a"d"/"dθ"(θ - sin θ)`
= a(1 – cos θ) ...(1)
and
`"dy"/"dθ" = a"d"/"dθ"(1 - cos θ)`
= a[0 – (– sin θ)]
= a sin θ
∴ `"dy"/"dx" = (("dy"/"dθ"))/(("dx"/"dθ")`
= `"a sin θ"/"a(1 - cos θ)"`
= `(2sin(θ/2).cos(θ/2))/(2sin^2(θ/2)) = cot(θ/2)`
and
`(d^2y)/(dx^2) = "d"/"dx"[cot(θ/2)]`
= `"d"/"dx"[cot(θ/2)].("d"θ/2)/"dx"]`
= `-"cosec"^2(θ/2)."d"/"dθ"(θ/2) xx (1)/(("dx"/"dθ")`
= `-"cosec"^2(θ/2) xx (1)/(2) xx (1)/(a(1 - cosθ)` ...[by (1)]
= `-(1)/(2a)"cosec"^2(θ/2) xx (1)/(2sin^2(θ/2)`
=`-(1)/(4a)."cosec"^4(θ/2)`.
APPEARS IN
संबंधित प्रश्न
Find dy/dx if x sin y + y sin x = 0.
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
x3 + x2y + xy2 + y3 = 81
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
if `x^y + y^x = a^b`then Find `dy/dx`
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `dy/dx if x^3 + y^2 + xy = 7`
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
Differentiate tan-1 (cot 2x) w.r.t.x.
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Find `"dy"/"dx"` if x = at2, y = 2at.
Find `"dy"/"dx"`, if : x = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.
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 xx w.r.t. xsix.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
Find the nth derivative of the following : (ax + b)m
Find the nth derivative of the following : cos (3 – 2x)
Find the nth derivative of the following:
y = e8x . cos (6x + 7)
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)` = .........
If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`
If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.
If y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.
Find `"dy"/"dx"` if, yex + xey = 1
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
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 x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
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 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^sqrt t`
