# How do you determine the derivative of this exponential function f(x) = sqrt (3^x) / x^2?

Write the function as $f \left(x\right) = \setminus \frac{{3}^{\frac{x}{2}}}{{x}^{2}}$ and then use the Quotient Rule and the Chain Rule:
$f ' \left(x\right) = \setminus \frac{{x}^{2} \setminus \cdot \ln \left(3\right) \setminus \cdot {3}^{\frac{x}{2}} \setminus \cdot \setminus \frac{1}{2} - {3}^{\frac{x}{2}} \setminus \cdot 2 x}{{x}^{4}} = \setminus \frac{{3}^{\frac{x}{2}} \setminus \cdot \left(\ln \left(\setminus \sqrt{3}\right) \setminus \cdot x - 2\right)}{{x}^{3}}$