证明dy/dx = (t/(t+2))^2 - 错误分析

To prove that dy/dx = (t/(t+2))^2, we need to find the derivative of y with respect to x and then simplify it to match the given expression.
\nFirst, let's find the derivative of y with respect to t:
dy/dt = d/dt (1/(t+2))
To differentiate 1/(t+2), we can use the chain rule:
dy/dt = (-1/(t+2)^2) * (d/dt (t+2))
dy/dt = (-1/(t+2)^2) * 1
dy/dt = -1/(t+2)^2
\nNow, let's find the derivative of x with respect to t:
dx/dt = d/dt (1/t)
dx/dt = -1/t^2
\nTo find dy/dx, we need to divide dy/dt by dx/dt:
dy/dx = (dy/dt) / (dx/dt)
dy/dx = (-1/(t+2)^2) / (-1/t^2)
dy/dx = (t^2 / (t+2)^2)
\nTo simplify this expression, let's multiply the numerator and denominator by (t+2)^2:
dy/dx = (t^2 / (t+2)^2) * ((t+2)^2 / (t+2)^2)
dy/dx = (t^2 / (t+2)^2) * (t^2 + 4t + 4) / (t+2)^2
dy/dx = (t^2 * (t^2 + 4t + 4)) / (t+2)^4
\nNow, let's simplify further:
dy/dx = (t^4 + 4t^3 + 4t^2) / (t+2)^4
\nWe can see that dy/dx is not equal to (t/(t+2))^2. Therefore, the statement dy/dx = (t/(t+2))^2 is not true.