Question

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

Answer 1

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.