Question

How do you differentiate `f(x)= e^x/(e^(x-2) +2x )` using the quotient rule?

Answer 1

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

We apply the Quotient Rule for Differentiation:

`d/dx(u/v) = (v(du)/dx-u(dv)/dx)/v^2`, or less formally, `" "(u/v)' = (v(du)-u(dv))/v^2`

I was taught to remember the rule in word; " vdu minus udv all over v squared ". To help with the ordering I was taught to remember the acronym, VDU as in Visual Display Unit.

So with `f(x) = e^x/(e^(x-2) +2x)` :

`{ ("Let, "u=e^x, => , (du)/dx=e^xx), ("And, "v=e^(x-2) +2x, =>, (dv)/dx=e^(x-2) +2 ) :}`

Then:

`f'(x) = ((e^(x-2) +2)(e^x) - (e^(x-2) +2x)(e^x))/(e^(x-2) +2x)^2`
`" " = e^x*(e^(x-2) +2 - e^(x-2) - 2x)/(e^(x-2) +2x)^2`
`" " = e^x*(2 - 2x)/(e^(x-2) +2x)^2`
`" " = (2e^x(1 - x))/(e^(x-2) +2x)^2`