If x(t)= e^2t and y(t)=sin(2t), then which of the following is an expression for dy/dx

Answer 1

dy/dx is the derivative of y(t) with respect to x(t).

First, we need to express y(t) in terms of x(t):
y(t) = sin(2t)
To find t in terms of x, we solve the equation x = e^(2t) for t:
x = e^(2t)
2t = ln(x)
t = ln(x)/2

Now, substitute t = ln(x)/2 into y(t):
y(ln(x)/2) = sin(2*ln(x)/2) = sin(ln(x))

dy/dx is the derivative of y(ln(x)/2) with respect to x:
dy/dx = (d(sin(ln(x))/dx)
= cos(ln(x)) * (d(ln(x))/dx)
= cos(ln(x)) * (1/x)
= cos(ln(x))/x

Therefore, the expression for dy/dx is cos(ln(x))/x.

Answer 2

or, as usual, dy/dx = (dy/dt)/(dx/dt)
= 2cos2t / 2e^2t = cos(2t)/e^(2t)

Answer 3

Yes, you are correct. Another way to express dy/dx is:

dy/dx = (dy/dt)/(dx/dt)
= (2cos2t)/(2e^(2t))
= cos(2t)/e^(2t)

Thank you for pointing out the alternative method.