Advertisements
Advertisements
Question
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
Advertisements
Solution
Let y = `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
Put x = tanθ. Thenθ = tan–1x
∴ `(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)) = sqrt(1 + tan^2θ + tanθ)/(sqrt(1 + tan^2θ) - tanθ)`
= `"secθ + tanθ"/"secθ - tanθ"`
= `((1/cosθ) + (sinθ/cosθ))/((1/cosθ) - (sinθ/cosθ))`
= `(1 + sinθ)/(1 - sinθ)`
= `(1 - cos(pi/2 + θ))/(1 + cos(pi/2 + θ)`
= `(2sin^2(pi/4 + θ/2))/(2cos^2(pi/4 + θ/2)`
= `tan^2(pi/4 + θ/2)`
∴ `sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x)`
= `tan(pi/4 + θ/2)`
∴ y = `tan^-1[tan(pi/4 + θ/2)]`
= `pi/4 + θ/2`
= `pi/4 + 1/2tan^-1x`
∴ `"d"/"dx"(pi/4) + (1)/(2)"d"/"dx"(tan^-1x)`
= `0 + (1)/(2) xx (1)/(1 + x^2)`
= `(1)/(2(1 + x^2)`.
APPEARS IN
RELATED QUESTIONS
Find dy/dx if x sin y + y sin x = 0.
Find `bb(dy/dx)` in the following:
sin2 y + cos xy = k
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`
Differentiate tan-1 (cot 2x) w.r.t.x.
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 = at2, y = 2at.
Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`
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 = a(θ – sin θ), y = a(1 – cos θ)
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 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
Find the nth derivative of the following : (ax + b)m
Find the nth derivative of the following : apx+q
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?
Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.
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 log (x + y) = log (xy) + a then show that, `"dy"/"dx" = (- "y"^2)/"x"^2`.
State whether the following is True or False:
The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
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)
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 y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.
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 y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
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`
