Advertisements
Advertisements
Question
If x = `(t + 1)/(t - 1), y = (t - 1)/(t + 1), "then show that" y^2 + "dy"/"dx"` = 0.
Solution
x = `("t" + 1)/("t" - 1)` .......(1)
y = `("t" - 1)/("t" + 1)` ......(2)
Multiplying equations (1) and (2)
x × y = `(("t" + 1))/(("t" - 1)) xx (("t" - 1))/(("t" + 1))`
x × y = 1
x = `1/"y"` ....(3)
Differentiating w.r.t. x,
`"x" xx "dy"/"dx" + "y" xx (1) = 0`
`1/"y" xx "dy"/"dx" + "y"/1 = 0` ....[From (3)]
`(1 xx "dy"/"dx" + "y"^2)/"y" = 0`
`"dy"/"dx" + "y"^2 = 0`
∴ `"y"^2 + "dy"/"dx" = 0`
Hence proved.
APPEARS IN
RELATED QUESTIONS
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:
`y = sin^(-1)((2x)/(1+x^2))`
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|?
If f (x) = |x − 2| write whether f' (2) exists or not.
Write the derivative of f (x) = |x|3 at x = 0.
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
Find `(dy)/(dx) if y = cos^-1 (√x)`
Differentiate tan-1 (cot 2x) w.r.t.x.
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
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 = cosec2θ, y = cot3θ at θ= `pi/(6)`
Find `dy/dx` if : x = 2 cos t + cos 2t, y = 2 sin t – sin 2t at t = `pi/(4)`
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
If x2 + 6xy + y2 = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.
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 = e8x . cos (6x + 7)
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^2[cot^-1(sqrt((1 + x)/(1 - x)))]`
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If `x = e^(x/y)`, then show that `"dy"/"dx" = (x - y)/(xlogx)`
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
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 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, x3 + x2y + xy2 + y3 = 81
If log (x + y) = log (xy) + a then show that, `"dy"/"dx" = (- "y"^2)/"x"^2`.
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"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 x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
Find `(dy)/(dx)`, if `y = sin^-1 ((2x)/(1 + x^2))`
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
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`
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 y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
Solve the following.
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`
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`
