Question

given y=e^x-e^-x/e^x+e^-x, prove that dy/dx=1-y^2

Answer 1

y=(e^x-e^-x)/(e^x+e^-x)=tanh(x), dy/dx=sech²(x)=1-tanh²(x)=1-y².

Without using hyperbolic functions we can still prove the DE.

y²=(e^2x-2+e^-2x)/(e^2x+2+e^-2x), 1-y²=4/(e^2x+2+e^-2x)=(2/(e^x+e^-x))².

Let u=e^x-e^-x, v=e^x+e^-x then du/dx=e^x+e^-x, dv/dx=e^x-e^-x,

y=u/v, dy/dx=(vdu/dx-udv/dx)/v².

dy/dx=[(e^x+e^-x)²-(e^x-e^-x)²]/(e^x+e^-x)²=(2/(e^x+e^-x))²=1-y² QED