How do you find the derivative of f(x)= x/(x-1)f(x)=xx1?

1 Answer
Dec 23, 2016

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

Explanation:

You could use the quotient rule, but I typically avoid doing this whenever possible as I find it tends to lead to higher chance of making an error and is generally more strenuous. To differentiate using the product rule, rewrite as

f(x)=x(x-1)^-1

Product rule:

f(x)=g(x)h(x)

f'(x)=g(x)h'(x) + g'(x)h(x)

In our case, g(x)=x and h(x)=(x-1)^-1

Leaving g(x) alone and multiply by the derivative of h(x), for which we would use the chain rule.

h'(x)=-(x-1)^-2*1

Where 1 is the derivative of the inside term, x-1.

Then, we leave h(x) alone and multiply by g'(x)

g'(x)=1

Putting it all together, we have

f'(x)=-x(x-1)^-2+(x-1)^-1

Which is equivalent to

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

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

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