Final answer:
The nth derivative of the function f(x) = 5e⁻³x is found by repeatedly applying the chain rule, which results in the general formula for the nth derivative as (-3)⁻³ * 5e⁻³x.
Step-by-step explanation:
To find the formula for the nth derivative of the function f(x) = 5e⁻³x, we begin by differentiating the function once. The first derivative of this function is f'(x) = -15e⁻³x, applying the chain rule by differentiating the exponent and multiplying by the derivative of the exponent with the original function.
We can generalize this pattern for higher order derivatives.
Each time we differentiate, we will multiply the result by -3 (due to the chain rule), meaning that the nth derivative will always involve raising -3 to the power of n. The nth derivative of f(x) is given by f(n)(x) = 5 * (-3)n * e⁻³x, which is equivalent to (-3)n * 5e⁻³x.