Final answer:
To find the derivative of f(x) = axe^{-bx}, we apply the product rule and the chain rule of differentiation, yielding -abxe^{-bx} as the answer.
Step-by-step explanation:
The question asks us to find the derivative of the function f(x) = axe^{-bx}. The correct answer is -abxe^{-bx}, which we find by applying the product rule and the chain rule of differentiation.
The product rule tells us that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Since e^{-bx} involves an exponential function and a chain of functions (-bx), the chain rule is also needed.
Let's go through the steps to differentiate f(x) = axe^{-bx}:
Differentiate ax, which gives us a.
Keep ax and differentiate e^{-bx} using the chain rule, resulting in -be^{-bx}.
Combine these results using the product rule: (d/dx)ax * e^{-bx} = a * e^{-bx} + ax * (-be^{-bx}).
Simplify the expression: -abxe^{-bx}.
Therefore, the derivative of f(x) = axe^{-bx} is -abxe^{-bx}.