Question

Y' = (e^x(1 + e^x) - e^(2x)) / (1 + e^x)²

Answer 1

To simplify the given expression, expand the numerator and denominator, and then substitute the simplified expressions back into the original equation we get y' = e^x / (1 + 2e^x + e^(2x)).

Step-by-step explanation:

To simplify the expression y' = (e^x(1 + e^x) - e^(2x)) / (1 + e^x)², we can expand the numerator and denominator.

Expanding the numerator, we have e^x + e^(2x) - e^(2x), which simplifies to e^x.

Expanding the denominator, we have (1 + e^x)(1 + e^x), which simplifies to 1 + 2e^x + e^(2x).

Now, we can substitute these simplified expressions back into the original equation y' = (e^x(1 + e^x) - e^(2x)) / (1 + e^x)², we get:

y' = e^x / (1 + 2e^x + e^(2x)).

y' = (e^x(1 + e^x) - e^(2x)) / (1 + e^x)². Simplify.