Question

Consider y = f(t) = ate^[-bt}, where a and b are positive constants. (a) Find f'(t) and confirm that it can be simplified as f'(t) = ae{-bt}(1 – bt).

Answer 1

The derivative of the function y = ate^{-bt} is found using the product rule of differentiation, which results in f'(t) = ae^{-bt}(1 - bt) after simplifying.

Step-by-step explanation:

The student has been asked to differentiate the function y = f(t) = ate^{-bt}, where a and b are positive constants. To find f'(t), we must apply the product rule of differentiation, which states 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.

For the function f(t) = ate^{-bt}, let us denote u(t) = at and v(t) = e^{-bt}. The product rule gives us:

f'(t) = u'(t)v(t) + u(t)v'(t)

Thus, we differentiate u(t) and v(t) separately:

• u'(t) = a
• v'(t) = -be^{-bt}

Substituting these into the product rule equation, we get:

f'(t) = a * e^{-bt} + at(-be^{-bt})

Simplifying:

f'(t) = ae^{-bt}(1 - bt)

Answer 2

To find the derivative f'(t) of the function y = f(t) = ate^{-bt}, we apply the product rule. The result is f'(t) = ae^{-bt}(1 - bt), confirming the required derivative.

Step-by-step explanation:

The given function is y = f(t) = ate^{-bt}, where a and b are positive constants. To find the derivative of the function, f'(t), we apply the product rule and the chain rule of differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function.

The function ate^{-bt} is the product of at and e^{-bt}. The derivative of at with respect to t is simply a, and the derivative of e^{-bt} with respect to t is -be^{-bt}. Applying the product rule, we get:

f'(t) = a · e^{-bt} + (-bt) · ate^{-bt}

f'(t) = ae^{-bt} - abte^{-bt}

f'(t) = ae^{-bt}(1 - bt)

This simplifies to the required derivative, confirming that f'(t) can indeed be simplified as ae^{-bt}(1 - bt).