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 ex is always ex, 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.