Question

How do you differentiate `x/e^(3x)`?

Answer 1

Apply the quotient rule to get `(1-3x)/(e^(3x))`.

Explanation:

The quotient rule says that, for two functions of `x`, call them `u` and `v`,
`d/dx(u/v)=(u'v-uv')/v^2`
In this problem, `u=x` and `v=e^(3x)`. Step 1 is to take the derivative of `u` and `v` to make the calculations easier:
`u=x`
`u'=1`

`v=e^(3x)`
`v'=3e^(3x)`
We can now make the substitutions:
`d/dx(x/e^(3x))=((x)'(e^(3x))-(x)(e^(3x))')/((e^(3x))^2)`
`d/dx(x/e^(3x))=(e^(3x)-3xe^(3x))/((e^(3x))^2)`

And finish off with a little algebra:
`d/dx(x/e^(3x))=(e^(3x)(1-3x))/((e^(3x))^2)`
`d/dx(x/e^(3x))=(1-3x)/(e^(3x))`