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Question Number 118798 by WasimShaikh last updated on 19/Oct/20

$$1)Find\frac{dy}{dx};ifx={at}^2,y=2at$$$$$$$$2)$$

Answered by Dwaipayan Shikari last updated on 20/Oct/20

$$x={at}^2$$$$\frac{dx}{dt}=2at$$$$y=2at$$$$\frac{dy}{dt}=2a\frac{dy}{dx}=\frac{dy}{dt}.\frac{dt}{dx}=\frac{1}{t}x={at}^{2}y=2at$$$$\frac{x}{y}=\frac{t}{2}\Rightarrow \frac{x}{2a}=\frac{t^2}{2}\Rightarrow t=\pm \sqrt{\frac{x}{a}}$$$$So$$$$\frac{dy}{dx}=\pm \sqrt{\frac{a}{x}}$$

Answered by Olaf last updated on 19/Oct/20

$$x={at}^2=a{\left(\frac{y}{2a}\right)}^2=\frac{y^2}{4a}$$$$y=\pm \sqrt{4ax}$$$$\frac{dy}{dx}=\pm \frac{4a}{2\sqrt{4ax}}=\pm \sqrt{\frac{a}{x}}$$

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