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Chapter 3: Problem 23

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

Short Answer

Expert verified
Question: Find the derivative of the function $$f(x) = xe^{-x}$$. Answer: The derivative of the given function is $$f'(x) = e^{-x} - xe^{-x}$$.

Step by step solution

01

Differentiate the individual functions

First, we'll find the derivatives of both functions $$u(x)$$ and $$v(x)$$. For $$u(x)=x$$: $$u'(x) = 1$$ For $$v(x)= e^{-x}$$: $$v'(x)=-e^{-x}$$
02

Apply the product rule

Now, we'll use the product rule to find the derivative of the function $$f(x)$$. $$f'(x) = u'(x)v(x) + u(x)v'(x) $$
03

Substitute the derivatives and simplify

We now substitute the expressions we found in step 1 into the formula for the derivative $$f'(x)$$. $$f'(x)= (1)(e^{-x}) + (x)(-e^{-x})$$ $$f'(x) = e^{-x} - xe^{-x} $$ So, the derivative of the function $$f(x) = xe^{-x}$$ is: $$f'(x) = e^{-x} - xe^{-x}$$

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