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Let there be two functions expressed in the form of a parametric variable, $y=f(t)$ and $x=g(t)$and I have find the second derivative of $y$ with respect to $x$.

To do that, I have done as shown $$\frac{d^2y}{dx^2}= \frac{d}{dt}(\frac{dy}{dt})×(\frac{dt}{dx})^2$$ $$\frac{d^2y}{dx^2} = \frac{d^2y}{dt^2} \biggm/\left(\frac{dx}{dt}\right)^2$$ But I am not getting the correct answer and I don't know what is the problem with this. I want to know if I have done something wrong in the above procedure?

Lalit Tolani
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Cyberax
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  • start with the correct expression for $\frac{dy}{dx}$ and differentiate with respect to $x$ using the Chain Rule – David Quinn Aug 17 '21 at 08:10
  • Thanks but i want. To why this one is not working – Cyberax Aug 18 '21 at 01:25
  • see the first solution to this https://math.stackexchange.com/questions/2675108/explanation-behind-second-derivative-of-a-parametric-equation-formula?rq=1 to see how it's supposed to be done – David Quinn Aug 18 '21 at 09:21
  • Thanks for it was helpful but i am now wondering that solving the given equation above from right hand side will give me same that is written in left hand side so why this is not working? – Cyberax Aug 19 '21 at 07:46
  • I have added the procedure too – Cyberax Aug 19 '21 at 07:55

1 Answers1

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The first expression of yours is wrong

$\displaystyle\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{d}{dx}(\frac{\frac{dy}{dt}}{\frac{dx}{dt}})=\frac{d}{dt}(\frac{\frac{dy}{dt}}{\frac{dx}{dt}})\frac{dt}{dx}$ which on evaluating by quotient rule gives

$$\displaystyle\frac{\frac{dx}{dt}.\frac{d^2y}{dt^2}-\frac{dy}{dt}\frac{d^2x}{dt^2}}{(\frac{dx}{dt})^3}$$

Lalit Tolani
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