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Chapter 3: Problem 9

Find the derivative of the following functions. $$f(t)=t^{5} e^{t}$$

Short Answer

Expert verified
Answer: The derivative of the function $$f(t)=t^{5} e^{t}$$ is $$f'(t) = e^{t}(5t^{4} + t^{5})$$.

Step by step solution

01

Identify the functions

First, identify the two functions in the given function: $$f(t)=t^{5} e^{t}$$ Here, we have: Function 1: $$u(t)=t^{5}$$ (a power function) Function 2: $$v(t)=e^{t}$$ (an exponential function)
02

Find the derivatives of the functions

Now that we've identified the functions, we need to find their derivatives using the Power Rule and the Exponential Rule. For Function 1: $$u(t)=t^{5}$$ Using the Power Rule: $$\frac{d}{dt}(t^{5})=5t^{4}$$ So, $$u'(t)=5t^{4}$$ For Function 2: $$v(t)=e^{t}$$ Using the Exponential Rule: $$\frac{d}{dt}(e^{t})=e^{t}$$ So, $$v'(t)=e^{t}$$
03

Apply the Product Rule to find the derivative of the given function

Now, we will apply the Product Rule to find the derivative of the given function $$f(t)=t^{5} e^{t}$$. Recall the Product Rule: $$(u\cdot v)'=u'\cdot v + u\cdot v'$$ Substitute the functions and their derivatives into the Product Rule: $$(t^{5} e^{t})'=(5t^{4}) \cdot (e^{t}) + (t^{5}) \cdot (e^{t})$$
04

Simplify the expression for the derivative

Simplify the expression obtained in Step 3 as much as possible: $$f'(t) = (5t^{4})\cdot(e^{t})+(t^{5})\cdot(e^{t})$$ Factor out the common term $$e^{t}$$: $$f'(t) = e^{t}(5t^{4} + t^{5})$$
05

Write the final answer

The derivative of the given function $$f(t)=t^{5} e^{t}$$ is: $$f'(t) = e^{t}(5t^{4} + t^{5})$$

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Most popular questions from this chapter

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