If x = at^2 and y = 2at, then frac{d^2y}{dx^2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $x = at^2$ and $y = 2at$.First, find $\frac{dx}{dt} = 2at$ and $\frac{dy}{dt} = 2a$.Then, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2a}{2

Vedclass · Accepted Answer

$-\frac{a}{xy}$

Vedclass · Answer

(A) Given $x = at^2$ and $y = 2at$.First, find $\frac{dx}{dt} = 2at$ and $\frac{dy}{dt} = 2a$.Then, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2a}{2at} = \frac{1}{t}$.Now, differentiate with respect to $x$:$\frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{1}{t}) = \frac{d}{dt}(\frac{1}{t}) \cdot \frac{dt}{dx} = (-\frac{1}{t^2}) \cdot \frac{1}{2at} = -\frac{1}{2at^3}$.We know $x = at^2$, so $t^2 = \frac{x}{a}$. Also $y = 2at$, so $t = \frac{y}{2a}$.Substitute $t^2 = \frac{x}{a}$ and $t = \frac{y}{2a}$ into the expression:$\frac{d^2y}{dx^2} = -\frac{1}{2a(t^2)(t)} = -\frac{1}{2a(\frac{x}{a})(\frac{y}{2a})} = -\frac{1}{\frac{xy}{a}} = -\frac{a}{xy}$.

If $x = at^2$ and $y = 2at$, then $\frac{d^2y}{dx^2} = \dots$

Similar Questions

The rate of change of $\sqrt{x^2+16}$ with respect to $\frac{x}{x-1}$ at $x=5$ is

If $x = a \cos^4 \theta$ and $y = a \sin^4 \theta$,then $\frac{dy}{dx}$ at $\theta = \frac{3\pi}{4}$ is

If $t$ is a parameter and $x=t+\frac{1}{t}, y=t-\frac{1}{t}$, then $\frac{d^2 y}{d x^2}=$

If $\tan y = \frac{2t}{1 - t^2}$ and $\sin x = \frac{2t}{1 + t^2},$ then $\frac{dy}{dx} = $

If $x = t^2$ and $y = t^3$,then $\frac{d^2y}{dx^2} =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module