**The derivative of e ^{-x} is -e^{-x}.** The derivative of e

^{-x}is found by applying the chain rule of derivatives and the knowledge that the derivative of e

^{x}is always e

^{x}, which can be found using a more complicated proof.

The chain rule of derivatives states that a composite function's derivative can be found by multiplying the inside function's derivative and the outside function's derivative. In this example, the larger function is e, and the inside function is -x. The outside function's derivative in this case is e^{-x}, and the inside function's derivative is -1. When multiplied, these give a resulting derivative of -e^{-x}.