Advertisements
Advertisements
Question
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
`x = (sin^3t)/sqrt(cos 2t), y = (cos^3t)/sqrt(cos 2t)`
Advertisements
Solution
Here x = `(sin^3t)/(sqrtcos 2t)` ....(1)
y = `(cos^3 t)/ (sqrtcos 2t)` .....(2)
Differentiating (1) and (2) w.r.t. t, we get,
`dx/dt = (sqrtcos2t d/dt sin^3 t - sin ^3 t d/dt (sqrt cos2t))/(cos2t)`
= `((sqrt cos2t) 3 sin^2 t cos t - sin^3 t. 1/(2 sqrtcos2t) . (-sin 2t).2)/(cos 2t)`
= `(sqrt cos 2t 3 sin^2 t cos t + (sin^3 t sin 2t)/(sqrtcos2t))/(cos 2t)`
= `(3 cos 2t sin^2 t cos t + sin^3 t sin 2t)/ ((cos 2t)^(3//2))`
`dy/dt = (sqrt cos 2t d/dt cos^3 t - cos^3 t d/dt sqrtcos2t)/(cos 2t)`
= `(sqrtcos2t.3 cos^2 t (- sint) - cos^3 t. 1/(2sqrtcos 2t).(-sin 2t).2)/(cos 2t)`
= `(-3 cos^2 t. sin t. sqrt cos2t + (cos^3 t sin 2t)/(sqrtcos2t))/(cos2t)`
= `(cos^3 t sin 2t - 3 cos^2 t. sin t cos 2t)/((cos2t)^(3//2))`
`dy/dx = (dy/dt)/(dx/dt)`
= `(cos^3 t sin 2t - 3 cos^2 t . sin t cos 2t)/(3 cos2t sin^2 t cos t + sin^3 t sin 2t)`
APPEARS IN
RELATED QUESTIONS
find dy/dx if x=e2t , y=`e^sqrtt`
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 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `at t = `pi/4`
If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find `dy/dx `
Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`
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.
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = 2at2, y = at4
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 = sin t, y = cos 2t
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 sec θ, y = b tan θ
If x = `sqrt(a^(sin^(-1)t))`, y = `sqrt(a^(cos^(-1)t))` show that `dy/dx = - y/x`.
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'?
If y = sin -1 `((8x)/(1 + 16x^2))`, find `(dy)/(dx)`
Evaluate : `int (sec^2 x)/(tan^2 x + 4)` dx
x = `"t" + 1/"t"`, y = `"t" - 1/"t"`
x = 3cosθ – 2cos3θ, y = 3sinθ – 2sin3θ
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)`
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 ______.
