Question

Question #64fc4

Answer 1

This will require the application of the chain rule.
First we see that `d/dx(blob)^2=2(blob)*(dblob)/dx` where blob is anything you can possibly imagine.
So we have `y'=2ln(1+e^x)*d/dx(ln(1+e^x))`

Next, `d/dx(ln(1+e^x))=1/(1+e^x) * e^x`
because the derivative of `e^x=e^x`

So putting everything together,

`y'=(2ln(1+e^x)*e^x)/(1+e^x)`

Finding y'' is more tricky because we will need to use quotient rule.
Remember, "Lowdeehi minus hideelow all over low squared."

`y''=[(1+e^x)*d/dx(2ln(1+e^x)*e^x)-2ln(1+e^x)*e^(2x)]/(1+e^x)^2`
`d/dx(2ln(1+e^x)*e^x)=2ln(1+e^x)*e^x+(2e^(2x))/(1+e^x)`

`={(1+e^x)[2ln(1+e^x)*e^x+(2e^(2x))/(1+e^x)]-2ln(1+e^x)*e^(2x)}/(1+e^x)^2`