Find dy/dx if y=cos2t, x=sint

Answer 1

To find dy/dx, we can use the chain rule. Let's start by expressing y in terms of x:

Since x = sin(t), we can solve for t in terms of x: t = arcsin(x).

Next, we can express y = cos(2t) in terms of t by substituting the value of t: y = cos(2arcsin(x)).

Now we differentiate y with respect to x using the chain rule:

dy/dx = (dy/dt)(dt/dx)

Differentiating y with respect to t, we get:

dy/dt = -2sin(2arcsin(x))

Differentiating t with respect to x, we get:

dt/dx = dt/dt * dt/dx = 1 * cos(t) = cos(arcsin(x))

Substituting these values back into the equation dy/dx = (dy/dt)(dt/dx), we have:

dy/dx = -2sin(2arcsin(x)) * cos(arcsin(x))

Simplifying further, we can rewrite sin(2arcsin(x)) in terms of x as 2x * √(1 - x^2):

dy/dx = -2 * 2x * √(1 - x^2) * cos(arcsin(x))

Finally, we can simplify this expression:

dy/dx = -4x * √(1 - x^2) * cos(arcsin(x))

Therefore, the derivative of y = cos(2t) with respect to x when x = sin(t) is given by dy/dx = -4x * √(1 - x^2) * cos(arcsin(x)).

Answer 2

And we have another botched attempt by the bot

y=cos2t, x=sint
dy/dt = -2 sin 2t , dx/dt = cos t

dy/dx = (dy/dt) / dx/dt = -2sin 2t/cos t
= -4 sint cost/cost
= -4sint

Answer 3

Apologies for the mistake in the previous response. You are correct. Let's correct the calculation:

Given:
y = cos(2t)
x = sin(t)

To find dy/dx, we can differentiate both sides of the equations with respect to t:

dy/dt = -2sin(2t)
dx/dt = cos(t)

Now, we can express dy/dx in terms of dy/dt and dx/dt:

dy/dx = (dy/dt) / (dx/dt)
= (-2sin(2t)) / cos(t)
= -2sin(2t) / cos(t)

Since we have x = sin(t), we can substitute 2t = arcsin(x):

dy/dx = -2sin(arcsin(x)) / cos(t)
= -2x / cos(t)
= -2x / √(1 - x^2) (using the trigonometric identity: cos(t) = √(1 - sin^2(t)) = √(1 - x^2))

Therefore, dy/dx = -2x / √(1 - x^2).

Answer 4

What a lot of work!

dy/dx = (dy/dt) / (dx/dt) = -2sin2t/cost = -4sint = -4x
note that this is the same answer as Bot came up with, since
cos(arcsin(x)) = √(1-x^2)

also, since y = cos2t = 1 - 2sin^2t = 1 - 2x^2
that means that dy/dx = -4x

Answer 5

Apologies for the confusion in the previous response. You are correct. The correct answer is indeed dy/dx = -4x. Thank you for catching that error.