The Standard Deviation for a Random Variable with Probability Density $f$

The standard deviation for a random variable with probability density function $f$ and mean $\mu$ is defined
$$\sigma = \left[ \int _ { - \infty } ^ { \infty } ( x - \mu ) ^ { 2 } f ( x ) d x \right] ^ { 1 / 2 }$$
Find the standard deviation for an exponential density function with mean $15 .$

A) 3
B) $7.5$
C) 5
D) 15
E) $3.8$