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

1 Answer
Jan 24, 2018

Using the quotient rule, given a function #f(x) = g(x)/(h(x))#, then the derivative #f'(x)" will be

#f'(x) = (g'(x)h(x) - g(x)h'(x))/(h^2(x))#. (1)

In this case, #g(x) = x# and #h(x) = e^(3-x) + x^3#. Then:

#g'(x)# = 1;

#h'(x) = -e^(3-x) + 3x^2;#

#[h(x)]^2 = (e^(3-x) + x^3)^2#.

Putting all these results into Equation (1):

#f'(x) = (e^(3-x) + x^3 - x(-e^(3-x) + 3x^2))/(e^(3-x) + x^3)^2#;

#f'(x) = (e^(3-x) + x^3 + xe^(3-x) - 3x^3)/(e^(3-x) + x^3)^2#;

#f'(x) = (e^(3-x)(x+1) - 2x^3)/(e^(3-x) + x^3)^2#.