By Thomas Bourdin • Updated Aug 30, 2022

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Understanding how functions change instantaneously is at the heart of calculus. The exponential function y = e^x is unique because it is its own derivative, making it a cornerstone of differential equations, growth models, and more. When the exponent is negative, we still use the same principles, but the process requires a slight twist.

Step 1: Identify the Function

Write down the function you want to differentiate. For this example, let y = e^-x.

Step 2: Apply the Chain Rule

The chain rule handles compositions of functions—here, the exponential function contains the linear function -x. In general:

y' = f'(g(x)) \times g'(x)

For y = e^g(x) with g(x) = -x, we have f'(g(x)) = e^g(x) and g'(x) = -1. Thus:

y' = e-x \times (-1) = -e-x

Step 3: Simplify the Result

Combining the terms gives the final derivative:

y' = -e^-x

This concise result shows that the slope of a negative exponential mirrors the original curve but points downward.