Advertisements
Advertisements
प्रश्न
If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`
Advertisements
उत्तर
Given that: x = 3sint – sin 3t, y = 3cost – cos 3t.
Differentiating both parametric functions w.r.t. t
`"dx"/"dt" = 3 cos "t" - cos 3"t" * 3`
= 3(cos t – cos 3t)
`"dy"/"dx" = -3 sin "t" + sin 3"t" * 3`
= 3(– sin t + sin 3t)
∴ `"dy"/"dx" = ("dy"/"dt")/("dx"/"dt")`
= `(3(- sin "t" + sin 3"t"))/(3(cos "t" - cos 3"t"))`
= `(-sin "t" + sin 3"t")/(cos "t" - cos 3"t")`
Put t = `pi/3`
`"dy"/"dx" = (- sin pi/3 + sin 3 (pi/3))/(cos pi/3 - cos 3 (pi/3))`
= `(- sqrt(3)/2 + sin pi)/(1/2 - cos pi)`
= `(- sqrt(3)/2 + 0)/(1/2 - (- 1))`
= `(- sqrt(3)/2)/(1/2 + 1)`
= `(- sqrt(3)/2)/(3/2)`
= `(-1)/sqrt(3)`
Hence, `"dy"/"dx" = (-1)/sqrt(3)`.
APPEARS IN
संबंधित प्रश्न
If `log_10((x^3-y^3)/(x^3+y^3))=2 "then show that" dy/dx = [-99x^2]/[101y^2]`
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=at2, y= 2at , then find dy/dx.
If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`
If `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`
If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1−cos 2t), show that `dy/dx=β/αtan t`
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`
Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`
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 = a (θ – sin θ), y = a (1 + cos 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'?
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
If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at t" = pi/4) = "b"/"a"`
Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan–1x, when x ≠ 0
If x = t2, y = t3, then `("d"^2"y")/("dx"^2)` is ______.
If `"x = a sin" theta "and y = b cos" theta, "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
If y `= "Ae"^(5"x") + "Be"^(-5"x") "x" "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
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 ______.
