Consider f(x)=x/(x^2+1). Find the derivative f'(x)

To answer this question we need to decide which differentiation rule to use . The quotient rule looks like the obvious choice, so lets try that. If f(x)=u(x)/v(x) (being careful that v(x) is not zero anywhere, so f makes sense) we have that f'(x)=(v(x)u'(x)-u(x)v'(x))/(v(x)^2). In our example we have u(x)=x and v(x)=x^2+1. Taking derivatives gives u'(x)=1 and v'(x)=2x. So applying the quotient rule we have f'(x)=((x^2+1)-2x^2)/((x^2+1)^2) = (1-x^2)/((x^2+1)^2)