Find `(dy)/(dx),` when `x=(e^t+e^(-t))/2a n dy(e^t-e^(-t))/2`

Explanation

`x=e^{t}, quad J=t cdot e^{-t}`

`frac{d x}{d t}=frac{d(e^{t})}{d t}=e^{t}`

`frac{d y}{d t}=frac{d(t cdot e^{-t})}{d t}=1 cdot e^{-t}+t cdot e^{-t}(-1)`

`=e^{-t}[1-t]`

`frac{d y}{d x}=frac{frac{d y}{d t}}{frac{d x}{d t}}=frac{bar{e}^{t(1-t)}}{e^{t}}=frac{1-t}{e^{2 t}}`

`therefore frac{d y}{d x}=frac{1-t}{e^{2 t}}=P` (say)

`frac{d^{2} y}{d x^{2}}=frac{d}{d x}(frac{d y}{d x})`

`=frac{d(p)}{d x}=frac{frac{d p}{d t}}{frac{d x}{d t}}`

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