Use the quotient Rule to differentiate the function f(x) = 8x/x^5+3
2 Answers
sciencesolve | Teacher | (Level 3) Educator Emeritus
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 | College Teacher | (Level 3) Valedictorian
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.