Use the quotient Rule to differentiate the function f(x) = 8x/x^5+3

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to use quotient rule to differentiate the function with respect to x, hence, you need to assign two functions to numerator and denominator, such that: `g(x) = 8x, h(x) = x^5 + 3` .

You need to remember the quotient rule, such that:

`f'(x) = (g'(x)h(x) - g(x)h'(x))/(h^2(x))`

Replacing 8 for g'(x) and `5x^4` for h'(x) yields:

`f'(x) = (8(x^5 + 3) - 8x(5x^4))/((x^5 + 3)^2)`

`f'(x) = (8x^5 + 24 - 40x^5)/((x^5 + 3)^2)`

`f'(x) = (24 - 32x^5)/((x^5 + 3)^2)`

`f'(x) = 8(3 - 4x^5)/((x^5 + 3)^2)`

Hence, evaluating the derivative of the given function, by quotient rule, yields `f'(x) = 8(3 - 4x^5)/((x^5 + 3)^2).`

llltkl | Student

According to the quotient rule the derivative of f(x)= `(g(x))/(h(x))` is given by:

`(g'(x)*h(x)-g(x)*h'(x))/[h(x)]^2`

The given function is f(x)=`(8x)/(x^5+3)` .

Differentiating using the quotient rule we get:

`f'(x)=((8x)'*(x^5+3)-(8x)*(x^5+3)')/(x^5+3)^2`

`=(8*(x^5+3)-8x*5x^4)/(x^5+3)^2`

`=(8(x^5+3-5x^5))/(x^5+3)^2`

`=(8(3-4x^5))/(x^5+3)^2`` rarr` answer.