Advertisements
Advertisements
प्रश्न
Differentiate `tan^-1((cosx)/(1 + sinx)) w.r.t. sec^-1 x.`
Advertisements
उत्तर
Let u = `tan^-1((cosx)/(1 + sinx)) and v = sec^-1x.`
Then we want to find `"du"/"dv"`.
Differentiate u and v w.r.t. x, we get
`"du"/"dx" = "d"/"dx"[tan^-1((cos)/(1 + sinx))]`
`(cosx)/(1 + sinx) = (sin(pi/2 - x))/(1 + cos(pi/2 - x)`
= `(2sin(pi/4 - x/2).cos(pi/4 - x/2))/(2cos^2(pi/4 - x/2)`
= `tan(pi/4 - x/2)`
∴ `"du"/"dx" = "d"/"dx"[tan^-1 tan(pi/4 - pi/2)]`
= `"d"/"dx"(pi/4 - x/2)`
= `"d"/"dx"(pi/4) - (1)/(2)"d"/"dx"(x)`
We know `d/dx(x) = 1` and the derivative of a constant is 0.
= `0 - (1)/(2) xx 1`
= `-(1)/(2)`
and
`"dv"/"dx" = "d"/"dx"(sec^-1x)`
= `(1)/(xsqrt(x^2 - 1)`
∴ `"du"/"dv" = (("du"/"dx"))/(("dv"/"dx")`
= `((-1/2))/((1/(xsqrt(x^2 - 1)))`
= `-(xsqrt(x^2 - 1))/(2)`.
APPEARS IN
संबंधित प्रश्न
Find `bb(dy/dx)` in the following:
xy + y2 = tan x + y
Find `bb(dy/dx)` in the following:
sin2 y + cos xy = k
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Find `dy/dx if x^3 + y^2 + xy = 7`
Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`
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 = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
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)
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 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 : sin (ax + b)
Find the nth derivative of the following : `(1)/(3x - 5)`
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)` = .........
Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`
Differentiate the following w.r.t. x : `sin^2[cot^-1(sqrt((1 + x)/(1 - x)))]`
If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
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 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
Find `"dy"/"dx"` if, xy = log (xy)
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.
If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
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 `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.
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 = 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`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
