Question

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

Answer 1

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`.