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

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

Question

(C) Given $x = at^2$ and $y = 2at$. First,differentiate $x$ with respect to $t$: $\frac{dx}{dt} = \frac{d}{dt}(at^2) = 2at$. Next,differentiate $y$ wi

Vedclass · Accepted Answer

$\frac{1}{t}$

Vedclass · Answer

(C) Given $x = at^2$ and $y = 2at$. First,differentiate $x$ with respect to $t$: $\frac{dx}{dt} = \frac{d}{dt}(at^2) = 2at$. Next,differentiate $y$ with respect to $t$: $\frac{dy}{dt} = \frac{d}{dt}(2at) = 2a$. Now,use the chain rule for parametric differentiation: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2a}{2at} = \frac{1}{t}$. Since $t = \frac{y}{2a}$,we can also express this as $\frac{dy}{dx} = \frac{2a}{y} = \frac{2a}{2at} = \frac{1}{t}$. Alternatively,from $y = 2at$,we have $t = \frac{y}{2a}$. Substituting $t$ into $x = at^2$: $x = a(\frac{y}{2a})^2 = a(\frac{y^2}{4a^2}) = \frac{y^2}{4a}$. $y^2 = 4ax$. Differentiating both sides with respect to $x$: $2y \frac{dy}{dx} = 4a$. $\frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y}$. Substituting $y = 2at$: $\frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t}$.

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

Similar Questions

If $x = \sin \theta$ and $y = \sin^3 \theta$,then the value of $\frac{d^2 y}{d x^2}$ at $\theta = \frac{\pi}{6}$ is:

Let $y = t^{10} + 1$ and $x = t^8 + 1,$ then $\frac{d^2y}{dx^2}$ is

If,for $a \neq 0$,$x = a(1 - \sin t)$ and $y = a(t + \cos t)$,then $\frac{d^2 y}{d x^2} = $

If $x=a \cos^{3} \theta$ and $y=a \sin^{3} \theta$,then $1+\left(\frac{dy}{dx}\right)^{2}$ is

Find the rate of change of $\sqrt{x^2 + 16}$ with respect to $\frac{x}{x - 1}$ at $x = 3$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module