# How do you differentiate  f(x)=x^7csc(x) using the product rule?

Oct 28, 2015

$f ' \left(x\right) = - {x}^{6} \csc \left(x\right) \left[x \cot \left(x\right) - 7\right]$

#### Explanation:

Keep the first factor and find the derivative of the second then add that to the derivative of the first factor and keep the 2nd factor untouched

$f \left(x\right) = u v$

$f ' \left(x\right) = u v ' + u ' v$

$f \left(x\right) = {x}^{7} \csc \left(x\right)$

$f ' \left(x\right) = {x}^{7} \left(- \csc \left(x\right) \cot \left(x\right)\right) + 7 {x}^{6} \csc \left(x\right)$

Factor out $- {x}^{6} \csc \left(x\right)$

$f ' \left(x\right) = - {x}^{6} \csc \left(x\right) \left[x \cot \left(x\right) - 7\right]$