Question

Use the Quotient Rule to differentiate the function. $$ f(x)=\frac{x}{x^{2}+1} $$

Short answer

\(f'(x) = \frac{1 - x^2}{(x^2 + 1)^2}\)

Step-by-step solution

Differentiate u(x) and v(x)

We first need to differentiate \(u(x)\) and \(v(x)\). The derivative of \(u(x) = x\) is \(u'(x) = 1\). The derivative of \(v(x)=x^{2}+1\) is \(v'(x) = 2x\).

Plug the derivatives into the Quotient Rule

Next, we plug \(u\), \(u'\), \(v\), and \(v'\) into the Quotient Rule \( (v*u' - u*v') / (v^2) \).\nSo we get \( ( (x^2 + 1) * 1 - x * 2x) / ((x^2 + 1)^2) = (x^2 + 1 - 2x^2) / ((x^2 + 1)^2) = (1 - x^2) / ((x^2 + 1)^2) \).

Simplify the result

We can simplify the derivative as \(f'(x) = \frac{1 - x^2}{(x^2 + 1)^2}\), which is the final answer.