# How do you differentiate f(x)= ( x - x tanx )/ (x -3)  using the quotient rule?

• Quotient rule: $\left(\frac{a}{b}\right) ' = \frac{a ' b - a b '}{b} ^ 2$
• We'll also need product rule for $x \tan x$, which consists on $\left(a b\right) ' = a ' b + a b '$
$\frac{f \left(x\right)}{\mathrm{dx}} = \frac{\left(1 - \left(\tan x + {\sec}^{2} x\right)\right) \left(x - 3\right) - \left(x - x \tan x\right)}{x - 3} ^ 2$
$\frac{f \left(x\right)}{\mathrm{dx}} = \frac{\cancel{x} - 3 - \cancel{x \tan x} + 3 \tan x - x {\sec}^{2} x + 3 {\sec}^{2} x - \cancel{x} + \cancel{x \tan x}}{x - 3} ^ 2$
$\frac{f \left(x\right)}{\mathrm{dx}} = \frac{3 \left(\tan x - 1\right) + \left(3 - x\right) {\sec}^{2} x}{x - 3} ^ 2$