Question

How do you differentiate `y= (2e^x) / (1+e^x)`?

Answer 1

`dy/dx=(2e^x)/(1+e^x)^2`

Explanation:

First of all, we see that the function is written in the format of `y=u/v`. This is equivalent to the quotient rule where:

`y=u/v`, `dy/dx=(((du)/dx)v-((dv)/dx)u)/v^2`

Now, separate the numerator and the denominator to differentiate. The easiest thing to do when differentiating is to take the constant term out (`2`) as shown below.

`dy/dx=(2e^x)/(1+e^x)`
`2(dy/dx)(e^x)/(1+e^x)`

`u=e^x`
`(du)/dx=e^x`

`v=1+e^x`
`(dv)/dx=e^x` (as the derivative of `1` is `0`)

Apply quotient rule:

`dy/dx=(((du)/dx)v-((dv)/dx)u)/v^2`
`dy/dx= ((e^x)(1+e^x)-(e^x)(e^x))/(1+e^x)^2`
`dy/dx=(e^x+e^2x-e^2x)/(1+e^x)^2`
`dy/dx=e^x/(1+e^x)^2`

Don't forget to multiply by the constant!

`dy/dx=2(e^x/(1+e^x)^2)`
`dy/dx=(2e^x)/(1+e^x)^2`