# How do you differentiate f(x)= (2 x^2 + 7 x - 2)/ (x sinx ) using the quotient rule?

The Quotient Rule says $\frac{d}{\mathrm{dx}} \left(f \frac{x}{g} \left(x\right)\right) = \frac{g \left(x\right) \cdot f ' \left(x\right) - f \left(x\right) \cdot g ' \left(x\right)}{{\left(g \left(x\right)\right)}^{2}}$. This, in addition to the product rule, linearity, and the power rule, gives:
$\frac{d}{\mathrm{dx}} \left(\frac{2 {x}^{2} + 7 x - 2}{x \sin \left(x\right)}\right) = \frac{x \sin \left(x\right) \cdot \left(4 x + 7\right) - \left(2 {x}^{2} + 7 x - 2\right) \cdot \left(\sin \left(x\right) + x \cos \left(x\right)\right)}{{x}^{2} {\sin}^{2} \left(x\right)}$
$= \frac{2 {x}^{2} \sin \left(x\right) + 2 \sin \left(x\right) - 2 {x}^{3} \cos \left(x\right) - 7 {x}^{2} \cos \left(x\right) + 2 x \cos \left(x\right)}{{x}^{2} {\sin}^{2} \left(x\right)}$