If y=at^2,x=2at,what is the value of d2y/dx^2 at x =1/2

1.	If y=at^2,x=2at,what is the value of d2y/dx^2 at x =1/2
Answer» ong>Answer: $\tt \dfrac{d^2y}{dx^2} =\dfrac{1}{2a}$ Step-by-step explanation: Given: y = at² x = 2at To Find: $\tt \dfrac{d^2y}{dx^2}\:\: at\:x=\dfrac{1}{2}$ SOLUTION: Given that y = at². Differentiating it with respect to t we get, $\tt \dfrac{dy}{dt} =\dfrac{d}{dt}( at^2)$ $\tt \dfrac{dy}{dt} =2at---(1)$ Also by given x = 2 at Differentiate with respect to t, $\tt \dfrac{dx}{dt} =\dfrac{d}{dt} (2at)$ $\tt \dfrac{dx}{dt} =2a---(2)$ Divide equation 1 by 2, $\tt \dfrac{dy/dt}{dx/dt} =\dfrac{2at}{2a}$ $\tt \dfrac{dy}{dx} =t$ Differentiating again with respect to x, $\tt \dfrac{d^2y}{dx^2} =\dfrac{d}{dx} \bigg(\dfrac{dy}{dx} \bigg)$ By chain rule, $\tt \dfrac{d^2y}{dx^2} =\dfrac{d}{dt} \bigg(\dfrac{dy}{dx} \bigg)\times \dfrac{dt}{dx}$ $\implies \tt \dfrac{d}{dt} \bigg(\dfrac{dy}{dx}\bigg) \div\dfrac{dx}{dt}$ $\implies \tt \dfrac{d}{dt} (t) \div 2a$ $\implies \tt \dfrac{1}{2a}$ SINCE this TERM is independent of x, at x = 1/2 the value would be 1/2a.

Answer»

ong>Answer:

$\tt \dfrac{d^2y}{dx^2} =\dfrac{1}{2a}$