How do you find the derivative of {e^{ - 3x}}?

To find the derivative of the function f(x) = e^(-3x), you can use the chain rule.

Here’s the step-by-step solution:

Recognize the outer function and the inner function:

The outer function is e^u, where u is the inner function.
The inner function is -3x.
Differentiate the outer function:

The derivative of e^u with respect to u is e^u.
Differentiate the inner function:

The derivative of -3x with respect to x is -3.
Apply the chain rule:

The chain rule tells us that the derivative of e^(-3x) is the derivative of the outer function (e^u) evaluated at the inner function, multiplied by the derivative of the inner function.
So, the derivative of e^(-3x) with respect to x is:

f'(x) = e^(-3x) * (-3)

Simplifying this gives:

f'(x) = -3e^(-3x)

Thus, the derivative of e^(-3x) is -3e^(-3x).