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Find a formula for the derivative of a given function

by RoRi

Find a formula for the derivative of the function

    \[ f(x) = \frac{x \sin x}{1 + x^2}. \]

First, we write,

    \[ f(x) = \frac{ x \sin x}{1+x^2} = x \left( \frac{\sin x}{1+x^2} \right). \]

Then, using the product and quotient rules for derivatives,

    \begin{align*} f'(x) &= \left( \frac{\sin x}{1+x^2} \right) + x \left( \frac{(1+x^2)\cos x - 2x \sin x}{(1+x^2)^2}\right) \\[8pt] &= \frac{\sin x + x \cos x }{1+x^2} - \frac{2x^2 \sin x}{(1+x^2)^2}. \end{align*}

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