Advertisements
Advertisements
प्रश्न
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)
Advertisements
उत्तर
Here x = a (cos θ + θ sin θ) ...(1)
y = y = a (sin θ – θ cos θ) ...(2)
Differentiating (1) and (2) w.r.t. θ, we get
`dx/(d θ)` = a [−sin θ + θ × cos θ + sin θ]
= a θ cos θ
`dy/(d theta)` = a [cos θ − (θ (−sin θ) + cos θ)]
= a [cos θ + θ sin θ − cos θ]
= a θ sin θ
`dy/dx = (dy/(dθ))/(dx/(dθ))`
= `(a θ sin θ)/(a θ cos θ)`
= tan θ
APPEARS IN
संबंधित प्रश्न
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 `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`
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), find `dy/dx `
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`
If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find `dy/dx `
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 = 2at2, y = at4
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 = cos θ – cos 2θ, y = sin θ – sin 2θ
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 = (sin^3t)/sqrt(cos 2t), y = (cos^3t)/sqrt(cos 2t)`
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 = `sqrt(a^(sin^(-1)t))`, y = `sqrt(a^(cos^(-1)t))` show that `dy/dx = - y/x`.
If `x = acos^3t`, `y = asin^3 t`,
Show that `(dy)/(dx) =- (y/x)^(1/3)`
x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`
x = 3cosθ – 2cos3θ, y = 3sinθ – 2sin3θ
If x = ecos2t and y = esin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`
If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`
Differentiate `x/sinx` w.r.t. sin x
If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0
If `"x = a sin" theta "and y = b cos" theta, "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
If x = `a[cosθ + logtan θ/2]`, y = asinθ then `(dy)/(dx)` = ______.
When \(x\) and \(y\) are expressed separately as functions of the same third variable \(t\), which equations are obtained?
In the parametric equations \(x=f(t)\) and \(y=g(t)\), what is \(t\) called?
Which formula is used after finding \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)?
If required, in terms of which variables may the final answer be expressed instead of the parameter?
For \(x=a\cos^3\theta\) and \(y=a\sin^3\theta\), which equation do these parametric equations represent?
The formula \(\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) is based on which rule?
