$Differentiate $y=((e^x)-(e^{-x}))/((e^x)+(e^{-x}))$ and prove that $dy/dx=1-y^2$$

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Differentiate $y=((e^x)-(e^{-x}))/((e^x)+(e^{-x}))$ and prove that $dy/dx=1-y^2$.

Differential Equations

Yap Guong Zer

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Mathe

I think there is a typo in the question.
Yap Guong Zer

There is no typo in the question, I'll try to rewrite to so that it is hopefully easier to understand basically it's e power of x minus e power of negative x over e power of x plus e power of negative x y=((e^x)-(e^-x))/((e^x)+(e^-x)) hope this helps
Mathe

if y = ((e^x)-(e^-x))/((e^x)+(e^-x)), it is not true that y = 1 - y^2. Consider the case when x = 0. In that case y = 0.
Yap Guong Zer

this is differentiation trying to prove that when y is differentiated, dy/dx=1-y^2. Are you saying that this is not achievable?
Mathe

So there was a typo, because you didn't include dy/dx before
Yap Guong Zer

ok, that was my bad , should have made it clearer, are you able to help with my quesrion now?

Yap Guong Zer

There is no typo in the question, I'll try to rewrite to so that it is hopefully easier to understand basically it's e power of x minus e power of negative x over e power of x plus e power of negative x y=((e^x)-(e^-x))/((e^x)+(e^-x)) hope this helps
Mathe

if y = ((e^x)-(e^-x))/((e^x)+(e^-x)), it is not true that y = 1 - y^2. Consider the case when x = 0. In that case y = 0.
Yap Guong Zer

this is differentiation trying to prove that when y is differentiated, dy/dx=1-y^2. Are you saying that this is not achievable?
Mathe

So there was a typo, because you didn't include dy/dx before
Yap Guong Zer

ok, that was my bad , should have made it clearer, are you able to help with my quesrion now?

Answer 1

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$y = \frac{e^x-e^{-x}}{e^x+e^{-x}} = \frac{e^x}{e^x}\cdot \frac{e^x-e^{-x}}{e^x+e^{-x}} = \frac{e^{2x}-1}{e^{2x}+1}$

$1-y^2 = 1- \frac{(e^{2x}-1)^2}{(e^{2x}+1)^2} = \frac{(e^{2x}+1)^2-(e^{2x}-1)^2}{(e^{2x}+1)^2} = \frac{e^{4x} +2e^{2x}+1 - (e^{4x} -2e^{2x}+1)}{(e^{2x}+1)^2} = \frac{4e^{2x}}{(e^{2x}+1)^2}$

$\frac{dy}{dx} = \frac{2e^{2x}\cdot(e^{2x}+1) - 2e^{2x}\cdot(e^{2x}-1)}{(e^{2x}+1)^2} = \frac{4e^{2x}}{(e^{2x}+1)^2}$

This shows that $\frac{dy}{dx} = 1-y^2$

Mathe

Yap Guong Zer

thank you very much

$Differentiate $y=((e^x)-(e^{-x}))/((e^x)+(e^{-x}))$ and prove that $dy/dx=1-y^2$$

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