Question

How do you find the derivative of `f(x)= -x/(x-1)`?

Answer 1

`f'(x)=x/(x-1)^2`

Explanation:

The most common way to evaluate this derivative is to use the quotient rule, but if we want to avoid using it to differentiate, we can be clever and use a bit of algebraic long division:
`-x/(x-1)=-(1+1/(x-1))=-(1+(x-1)^-1)`

We can then differentiate this expression using only the chain rule and letting `u=x-1`:
`-d/dx(1+(x-1)^-1)=-d/(du)(1+u^-1)d/dx(x-1)`

`=u^-2*x` (using the power rule)

`=x(x-1)^-2=x/(x-1)^2` (resubstitute and simplify)