Advertisements
Advertisements
Question
If x = `sqrt(a^(sin^(-1)t))`, y = `sqrt(a^(cos^(-1)t))` show that `dy/dx = - y/x`.
Advertisements
Solution
Given, x = `sqrt(a^(sin^(-1)t))` and y = `sqrt(a^(cos^(-1)t))`
`dx/dt = 1/2 . 1/(sqrt(a^(sin^(-1)t))). d/dt a ^(sin^(-1)t)`
= `1/2 . 1/ sqrt (a^(sin^(-1)t)). a^(sin^(-1)t) . log a d/dt sin^-1 t`
= `sqrt(a^(sin^(-1)t))/2. log a . 1/ (sqrt(1-t^2)`
`dy/dt = 1/2. 1/ sqrt (a^(cos^(-1)t)). d/dt a^(cos^(-1)t)`
= `1/2 . 1/sqrt (a^(cos^(-1)t)). a^( cos^(-1)t) . log a. (-1)/(sqrt (1 - t^2))`
= `sqrt (a^(cos^(-1)t))/2 .log a (-1)/sqrt(1 - t^2)`
`∴ dy/dx = (dy/dt)/(dx/dt)`
= `(sqrt (a^(cos^(-1)t))/2. log a. (-1)/ sqrt(1 - t^2))/( sqrt (a^(sin^(-1)t))/2 . log a . 1/ sqrt (1 - t^2))`
= `(-sqrt( a^(cos^(-1)t)))/ sqrt (a^(sin^(-1)t))`
= `(-y)/x`
APPEARS IN
RELATED QUESTIONS
If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and hence, find dy/dx if x=a cost, y=a sint
If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`
If y =1 − cos θ, x = 1 − sin θ, then `dy/dx "at" θ =pi/4` is ______
If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find `dy/dx `
If x = cos t (3 – 2 cos2 t) and y = sin t (3 – 2 sin2 t), find the value of dx/dy at t =4/π.
Derivatives of tan3θ with respect to sec3θ at θ=π/3 is
(A)` 3/2`
(B) `sqrt3/2`
(C) `1/2`
(D) `-sqrt3/2`
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = a cos θ, y = b cos θ
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = 4t, y = `4/y`
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = `a(cos t + log tan t/2)`, y = a sin t
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = a sec θ, y = b tan θ
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)
If `x = acos^3t`, `y = asin^3 t`,
Show that `(dy)/(dx) =- (y/x)^(1/3)`
IF `y = e^(sin-1x) and z =e^(-cos-1x),` prove that `dy/dz = e^x//2`
The cost C of producing x articles is given as C = x3-16x2 + 47x. For what values of x, with the average cost is decreasing'?
Evaluate : `int (sec^2 x)/(tan^2 x + 4)` dx
sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`
If x = ecos2t and y = esin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`
If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at t" = pi/4) = "b"/"a"`
If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`
Differentiate `x/sinx` w.r.t. sin x
Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan–1x, when x ≠ 0
If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0
Derivative of x2 w.r.t. x3 is ______.
If y `= "Ae"^(5"x") + "Be"^(-5"x") "x" "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
Form the point of intersection (P) of lines given by x2 – y2 – 2x + 2y = 0, points A, B, C, Dare taken on the lines at a distance of `2sqrt(2)` units to form a quadrilateral whose area is A1 and the area of the quadrilateral formed by joining the circumcentres of ΔPAB, ΔPBC, ΔPCD, ΔPDA is A2, then `A_1/A_2` equals
If x = `a[cosθ + logtan θ/2]`, y = asinθ then `(dy)/(dx)` = ______.
Let a function y = f(x) is defined by x = eθsinθ and y = θesinθ, where θ is a real parameter, then value of `lim_(θ→0)`f'(x) is ______.
