Advertisements
Advertisements
प्रश्न
DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.
Advertisements
उत्तर
Let u = `tan^-1((sqrt(1 + x^2) - 1)/(x))`
and
v = `tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`
Then we want to find `"du"/"dv"`
u = `tan^-1((sqrt(1 + x^2) - 1)/(x))`
Put x = tanθ
Then θ = tan–1 x
and
`(sqrt(1 + x^2) - 1)/(x) = (sqrt(1 + tan^2θ) - 1)/tanθ`
= `(secθ - 1)/(tanθ)`
= `((1)/(cosθ) - 1)/((sinθ/cosθ)`
= `(1 - cosθ)/(sinθ)`
= `(2sin^2(θ/2))/(2sin(θ/2)cos(θ/2))`
= `tan(θ/2)`
∴ u = `tan^-1[tan(θ/2)] = θ/(2) = (1)/(2)tan^-1x`
∴ `"du"/"dx" = (1)/(2)"d"/"dx"(tan^-1x)`
= `(1)/(2) xx (1)/(1 + x^2)`
= `(1)/(2(1 + x^2)`
v = `tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`
Put x = sin θ.
Then θ = sin–1x
and
`(2xsqrt(1 - x^2))/(1 - 2x^2)`
= `(2sinθsqrt(1 - sin^2θ))/(1 - 2sin^2θ)`
= `(2sinθcosθ)/(1 - 2sin^2θ)`
= `(sin2θ)/(cos2θ)`
= tan 2θ
∴ v = tan−1(tan2θ)
= 2θ
= 2sin−1x
∴ `"dv"/"dx" = 2"d"/"dx"(sin^-1x)`
= `2 xx (1)/sqrt(1 - x^2) = (2)/sqrt(1 - x^2)`
∴ `"du"/"dx" = (("du"/"dx"))/(("dv"/"dx")`
= `([(1)/(2(1 + x^2))])/(((2)/sqrt(1 - x^2))`
= `(1)/(2(1 + x^2)) xx sqrt(1 - x^2)/(2)`
= `sqrt(1 - x^2)/(4(1 + x^2)`
APPEARS IN
संबंधित प्रश्न
If y=eax ,show that `xdy/dx=ylogy`
Find `bb(dy/dx)` in the following:
2x + 3y = sin y
Find `bb(dy/dx)` in the following:
xy + y2 = tan x + y
Find `bb(dy/dx)` in the following:
x2 + xy + y2 = 100
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Show that the derivative of the function f given by
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
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)`.
Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`
Differentiate xx w.r.t. xsix.
Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)
Find the nth derivative of the following : cos x
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 : y = eax . cos (bx + c)
Choose the correct option from the given alternatives :
If y = sin (2sin–1 x), then dx = ........
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"` = ........
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))`
If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.
Find `"dy"/"dx"` if, yex + xey = 1
State whether the following is True or False:
The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`
If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`
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 x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
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 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 + sin(x + y) = y – cos(x – y)
Find `(d^2y)/(dy^2)`, if y = e4x
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 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`
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 = e3t, 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`
