How do you differentiate #f(x)= 1/(e^(3x) -5x)# using the quotient rule?

1 Answer
Mar 3, 2018

#(5-3e^(3x))/( e^(3x) - 5x)^2#

Explanation:

The quotient Rule states that
the derivative of a division of two functions #f(x)/g(x)#
Is equal to #(f'(x)g(x) - f(x)g'(x))/ (g(x))^2#

therefore in your Question,
let #f(x) = 1#
and #g(x) = e^(3x) - 5x#

their respective derivatives are
#f'(x) = 0#
#g'(x) = 3e^(3x) - 5#

therefore, the derivative of the entire equation using the Quotient rule is
#((0*e^(3x) - 5x) - (1*(3e^(3x) - 5)))/ ( e^(3x) - 5x)^2#

# = - (3e^(3x) - 5)/( e^(3x) - 5x)^2#

which is equal to
#(5-3e^(3x))/( e^(3x) - 5x)^2#
=
and you can simplify it more if you want but that's basically it