Question

How do you differentiate: `y=(x^4)(e^x)`?

Answer 1

Use the product rule, together with the fact that `d/dx(e^x) = e^x`

I use the product rule in the order: `(fg)' = f'g+fg'`

`y=x^4e^x`

`y' = 4x^3e^x+x^4e^x`

(Yes, I know its hard to tell where I took the derivative of `e^x` I did it in the second term.)

Here it is again with derivatives in `color(red)"red"`

`y=x^4e^x`

`y' = color(red)(4x^3)e^x+x^4color(red)(e^x)`

Of course we can factor if it seems like a good idea:

`y'=x^3e^x(4+x)`