How do you differentiate f(x)=x^6/e^(x-6) using the quotient rule?

Oct 30, 2015

$f ' \left(x\right) = \frac{{x}^{5} \left(6 - x\right)}{e} ^ \left(x - 6\right)$

Explanation:

$f ' \left(x\right) = \frac{u ' v - u v '}{v} ^ 2$

$u = {x}^{6} \implies u ' = 6 {x}^{5}$
$v = {e}^{x - 6} \implies v ' = {e}^{x - 6} \cdot \left(x - 6\right) ' = {e}^{x - 6} \cdot 1 = {e}^{x - 6}$

$f ' \left(x\right) = \frac{6 {x}^{5} {e}^{x - 6} - {x}^{6} {e}^{x - 6}}{{e}^{x - 6}} ^ 2 = \frac{{x}^{5} {e}^{x - 6} \left(6 - x\right)}{{e}^{x - 6}} ^ 2$

$f ' \left(x\right) = \frac{{x}^{5} \left(6 - x\right)}{e} ^ \left(x - 6\right)$