Advertisements
Advertisements
Question
Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)
Solution
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
RELATED QUESTIONS
If y=eax ,show that `xdy/dx=ylogy`
Find dy/dx if x sin y + y sin x = 0.
Find `dy/dx` in the following:
ax + by2 = cos y
Find `dy/dx` in the following:
xy + y2 = tan x + y
Find `dy/dx` in the following.
x3 + x2y + xy2 + y3 = 81
if `x^y + y^x = a^b`then Find `dy/dx`
Show that the derivative of the function f given by
If f (x) = |x − 2| write whether f' (2) exists or not.
Write the derivative of f (x) = |x|3 at x = 0.
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 = sin3θ , y = cos3θ
Differentiate e4x + 5 w.r..t.e3x
Differentiate tan-1 (cot 2x) w.r.t.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 = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
If x = `(t + 1)/(t - 1), y = (t - 1)/(t + 1), "then show that" y^2 + "dy"/"dx"` = 0.
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
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))`.
Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`
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:
`(1)/x`
Find the nth derivative of the following : apx+q
Find the nth derivative of the following : cos (3 – 2x)
Find the nth derivative of the following : y = eax . cos (bx + c)
Choose the correct option from the given alternatives :
Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is
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
Solve the following :
f(x) = –x, for – 2 ≤ x < 0
= 2x, for 0 ≤ x < 2
= `(18 - x)/(4)`, for 2 < x ≤ 7
g(x) = 6 – 3x, for 0 ≤ x < 2
= `(2x - 4)/(3)`, for 2 < x ≤ 7
Let u (x) = f[g(x)], v(x) = g[f(x)] and w(x) = g[g(x)]. Find each derivative at x = 1, if it exists i.e. find u'(1), v' (1) and w'(1). If it doesn't exist, then explain why?
Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1:
| x | f(x) | g(x) | f')x) | g'(x) |
| 0 | 1 | 5 | `(1)/(3)` | |
| 1 | 3 | – 4 | `-(1)/(3)` | `-(8)/(3)` |
(i) The derivative of f[g(x)] w.r.t. x at x = 0 is ......
(ii) The derivative of g[f(x)] w.r.t. x at x = 0 is ......
(iii) The value of `["d"/"dx"[x^(10) + f(x)]^(-2)]_(x = 1_` is ........
(iv) The derivative of f[(x + g(x))] w.r.t. x at x = 0 is ...
Differentiate the following w.r.t. x : `sin[2tan^-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:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
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 y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
If y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.
Find `"dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"`
Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0
Solve the following:
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` then `"dy"/"dx"` = ?
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
If `"x"^7*"y"^9 = ("x + y")^16`, then show that `"dy"/"dx" = "y"/"x"`
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
`(dy)/(dx)` of `2x + 3y = sin x` is:-
y = `e^(x3)`
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.
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`
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`.
