How do you differentiate x/e^(3x)xe3x?

1 Answer
Mar 13, 2016

Apply the quotient rule to get (1-3x)/(e^(3x))13xe3x.

Explanation:

The quotient rule says that, for two functions of xx, call them uu and vv,
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))