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

1 Answer
Ken C.
Mar 13, 2016

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))#

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