# How do you differentiate  (xsinx)/(x^2+1)?

Dec 12, 2016

We'll be using the product rule with the quotient rule:

Let's think of your function as:

$\frac{f \left(x\right) \cdot g \left(x\right)}{h \left(x\right)}$

Where:
$f \left(x\right) = x$
$g \left(x\right) = \sin \left(x\right)$
$h \left(x\right) = {x}^{2} + 1$

Thus, the derivative is:

$\frac{\frac{d}{\mathrm{dx}} \left(f \left(x\right) \cdot g \left(x\right)\right) \cdot h \left(x\right) - \left(f \left(x\right) \cdot g \left(x\right)\right) \cdot \frac{d}{\mathrm{dx}} \left(h \left(x\right)\right)}{h {\left(x\right)}^{2}}$

Plug in our functions and find the derivatives:

$\frac{\frac{d}{\mathrm{dx}} \left(x \sin x\right) \cdot \left({x}^{2} + 1\right) - \left(x \sin x\right) \cdot \frac{d}{\mathrm{dx}} \left({x}^{2} + 1\right)}{{\left({x}^{2} + 1\right)}^{2}}$

$= \frac{\left(\sin x + x \cos x\right) \left({x}^{2} + 1\right) - \left(2 {x}^{2} \sin x\right)}{{x}^{2} + 1} ^ 2$