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If x = (1 – t2 )/(1 + t2) y = 2t/(1 + t2) then dy/dx at t =1 is _____________.
- a)1/2
- b)1
- c)0
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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If x = (1 – t2 )/(1 + t2) y = 2t/(1 + t2) then dy/dx at t =1 is ...
We are given the following expressions:
x = (1 - t^2)/(1 + t^2)
y = 2t/(1 + t^2)
To find dy/dx at t = 1, we need to find the derivative of y with respect to x.
Derivative of y with respect to t can be found as follows:
dy/dt = (d/dt)(2t/(1 + t^2))
= (2(1 + t^2) - 2t(2t))/(1 + t^2)^2
= (2 + 2t^2 - 4t^2)/(1 + t^2)^2
= (2 - 2t^2)/(1 + t^2)^2
Now, to find dy/dx, we can use the chain rule:
dy/dx = (dy/dt)/(dx/dt)
To find dx/dt, we can differentiate x with respect to t:
dx/dt = (d/dt)((1 - t^2)/(1 + t^2))
= (-2t(1 + t^2) - 2(1 - t^2))/(1 + t^2)^2
= (-2t - 2t^3 - 2 + 2t^2)/(1 + t^2)^2
= (-2 + 2t^2 - 2t - 2t^3)/(1 + t^2)^2
Now, substitute dy/dt and dx/dt into the equation for dy/dx:
dy/dx = (2 - 2t^2)/(1 + t^2)^2 / (-2 + 2t^2 - 2t - 2t^3)/(1 + t^2)^2
= (2 - 2t^2)/(-2 + 2t^2 - 2t - 2t^3)
= (2 - 2t^2)/(-2(1 - t^2 + t + t^3))
= (2 - 2t^2)/(-2(1 + t)(1 - t + t^2))
Simplifying further:
dy/dx = (2 - 2t^2)/(-2(1 + t)(1 - t + t^2))
= (1 - t^2)/((1 + t)(t^2 - t + 1))
Now, substitute t = 1 into the equation:
dy/dx = (1 - 1^2)/((1 + 1)(1^2 - 1 + 1))
= 0/(2(1))
= 0
Therefore, the value of dy/dx at t = 1 is 0. Hence, the correct answer is option 'C'.
x = (1 - t^2)/(1 + t^2)
y = 2t/(1 + t^2)
To find dy/dx at t = 1, we need to find the derivative of y with respect to x.
Derivative of y with respect to t can be found as follows:
dy/dt = (d/dt)(2t/(1 + t^2))
= (2(1 + t^2) - 2t(2t))/(1 + t^2)^2
= (2 + 2t^2 - 4t^2)/(1 + t^2)^2
= (2 - 2t^2)/(1 + t^2)^2
Now, to find dy/dx, we can use the chain rule:
dy/dx = (dy/dt)/(dx/dt)
To find dx/dt, we can differentiate x with respect to t:
dx/dt = (d/dt)((1 - t^2)/(1 + t^2))
= (-2t(1 + t^2) - 2(1 - t^2))/(1 + t^2)^2
= (-2t - 2t^3 - 2 + 2t^2)/(1 + t^2)^2
= (-2 + 2t^2 - 2t - 2t^3)/(1 + t^2)^2
Now, substitute dy/dt and dx/dt into the equation for dy/dx:
dy/dx = (2 - 2t^2)/(1 + t^2)^2 / (-2 + 2t^2 - 2t - 2t^3)/(1 + t^2)^2
= (2 - 2t^2)/(-2 + 2t^2 - 2t - 2t^3)
= (2 - 2t^2)/(-2(1 - t^2 + t + t^3))
= (2 - 2t^2)/(-2(1 + t)(1 - t + t^2))
Simplifying further:
dy/dx = (2 - 2t^2)/(-2(1 + t)(1 - t + t^2))
= (1 - t^2)/((1 + t)(t^2 - t + 1))
Now, substitute t = 1 into the equation:
dy/dx = (1 - 1^2)/((1 + 1)(1^2 - 1 + 1))
= 0/(2(1))
= 0
Therefore, the value of dy/dx at t = 1 is 0. Hence, the correct answer is option 'C'.
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