What is d 2 y d x 2 \frac{d^2y}{dx^2} ?

Calculus Level 3

If x = a t 2 x = at^2 and y = 2 a t y = 2at , then what is d 2 y d x 2 \dfrac{d^2y}{dx^2} ?

− 1 t 2 -\frac{1}{t^2} 1 y t \frac{1}{yt} − 1 2 a t 3 -\frac{1}{2at^3} 1 4 t a 3 \frac{1}{4ta^3} 1 2 a t 2 \frac{1}{2at^2} − 1 4 t a 3 -\frac{1}{4ta^3}

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Munem Shahriar by Munem Shahriar , 18, Bangladesh

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2 solutions

d y d x = d y d t × d t d x = 2 a × 1 2 a t = 1 t \begin{aligned} \frac {dy}{dx} & = \frac {dy}{dt} \times \frac {dt}{dx} = 2a \times \frac 1{2at} = \frac 1t \end{aligned}

d 2 y d x 2 = d d x ( d y d x ) = d d x ( 1 t ) = d d t ( 1 t ) × d t d x = − 1 t 2 × 1 2 a t = − 1 2 a t 3 \begin{aligned} \frac {d^2y}{dx^2} & = \frac d{dx} \left(\frac {dy}{dx} \right) = \frac d{dx} \left(\frac 1t \right) = \frac d{dt} \left(\frac 1t \right) \times \frac {dt}{dx} = - \frac 1{t^2} \times \frac 1{2at} = \boxed{- \dfrac 1{2at^3}} \end{aligned}

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Aareyan Manzoor
Jun 4, 2017

alternatively: y = 2 a x → y ′ = a a x → y ′ ′ = − a 2 2 a x 3 = − a 2 2 ( a t ) 3 = − 1 2 a t 3 y=2\sqrt{ax}\to y'=\dfrac{a}{\sqrt{ax}}\to y''=\dfrac{-a^2}{2\sqrt{ax}^3}=\dfrac{-a^2}{2 (at)^3}=\dfrac{-1}{2at^3}

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