Question

How do you find the derivative of `ln(e^x+1)`?

Answer 1

`e^x/(e^x+1`

Explanation:

differentiate using the `color(blue)"chain rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))........ (A)`

let `color(blue)(u=e^x+1)rArr(du)/(dx)=e^x`

and `y=lncolor(blue)(u)rArr(dy)/(du)=1/color(blue)(u)`

substitute these values into (A) and convert `color(blue)(u)` back into x

`rArrdy/dx=1/color(blue)(u).e^x=e^x/(e^x+1)`