I'm attempting to find the derivative of the following function:
$$\frac{e^x – e^{-x}}{e^x + e^{-x}}$$
where I would use the quotient rule to find its derivative. In the end, I would obtain:
$$\frac{(e^x + e^{-x})^2 -(e^x – e^{-x})^2}{(e^x + e^{-x})^2}$$
However, I checked with the calculator online and I found out that the equation I current have above can be further simplified to:
$$\text{ Rewrite / Simplify }$$
$$=1-\frac{(e^x – e^{-x})^2}{(e^x + e^{-x})^2}$$
$$\text{ Simplify }$$
$$\frac{4e^{2x}}{(e^{2x} +1)^2}$$
I'm having issues understanding the simplification step above.
My biggest question is, in the second image above, why couldn't I have $(e^x + e^{-x})^2$ cancel out $(e^x + e^{-x})^2$ at the denominator, which would leave me with just $-(e^x – e^{-x})^2$? Could anyone explain the simplification step to me as I am very much confused.
P.S. pardon me but I couldn't understand the syntax on writing equations here so I would opt to use screenshot instead.
Best Answer
I think your issue is that the computation in the last "cartridge" should be decomposed into $2$ steps :
First step :
The first expression can be written, taking a common denominator :
$$\dfrac{(e^x+e^{-x})^2}{(e^x+e^{-x})^2}+\dfrac{(e^x-e^{-x})^2}{(e^x+e^{-x})^2}=\dfrac{(e^x+e^{-x})^2+(e^x-e^{-x})^2}{(e^x+e^{-x})^2}=\dfrac{4}{(e^x+e^{-x})^2}$$
Second step : Factor $e^{-x}$ in the denominator :
$$(e^x+e^{-x})^2=(e^{-x}(e^{2x}-1))^2=e^{-2x}(e^{2x}-1)^2$$
We have obtained a $(e^{2x}-1)^2$ as a factor in the denominator.
Getting rid of $e^{-2x}$ in the denominator by transforming it into $e^{2x}$ in the numerator, explains the final result with $4e^{2x}$ in the numerator.