# How do you differentiate #f(x) = (x)/(x^4-x^3+6)# using the quotient rule?

##### 1 Answer

Feb 7, 2016

#### Explanation:

The quotient rule states that

#d/dx[(g(x))/(h(x))]=(g'(x)h(x)-g(x)h'(x))/[h(x)]^2#

Applying this to the given function, we see that

#f'(x)=((x^4-x^3+6)d/dx[x]-xd/dx[x^4-x^3+6])/(x^4-x^3+6)^2#

Both of these derivatives can be found through the power rule:

#f'(x)=((x^4-x^3+6)(1)-(x)(4x^3-3x^2))/(x^4-x^3+6)^2#

Simplified, this gives

#f'(x)=(x^4-x^3+6-4x^4+3x^3)/(x^4-x^3+6)^2#

#f'(x)=(-3x^4+2x^3+6)/(x^4-x^3+6)^2#