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Magazine Chapter 3: Problem 32
Find \(d y\). $$y=x e^{-x}$$
Short Answer
Expert verified
The derivative is \(\frac{dy}{dx} = e^{-x}(1 - x).\)
Step by step solution
01
Apply Product Rule
The function given is a product of two functions: \(y = x \cdot e^{-x}\). To differentiate this, we'll use the product rule which is stated as \((u \, v)' = u' \, v + u \, v'\), where \(u = x\) and \(v = e^{-x}\).
02
Differentiate \(u = x\)
Differentiate \(x\) with respect to \(x\). The derivative is: \[u' = \frac{d}{dx}(x) = 1.\]
03
Differentiate \(v = e^{-x}\)
Differentiate \(e^{-x}\) with respect to \(x\). By applying the chain rule, we have: \[v' = \frac{d}{dx}(e^{-x}) = -e^{-x}.\]
04
Substitute into Product Rule Formula
Substitute the derivatives into the product rule formula: \[\frac{d}{dx}(x \cdot e^{-x}) = (x' \cdot e^{-x}) + (x \cdot (e^{-x})').\]Substitute the values: \[= (1 \cdot e^{-x}) + (x \cdot -e^{-x}).\]
05
Simplify the Expression
Simplify the derivative expression from Step 4: \[= e^{-x} - x e^{-x}.\]Combine the terms as they have a common factor, \(e^{-x}\):\[= e^{-x}(1 - x).\]
06
Finalize and Write the Result
The derivative of \(y = x \cdot e^{-x}\) with respect to \(x\) is: \[\frac{dy}{dx} = e^{-x}(1 - x).\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Product Rule
The product rule is a fundamental rule in calculus used when differentiating a product of two functions. It helps us find the derivative of expressions that multiply two different variables or functions together.
Consider the product rule formula: \((u \, v)' = u' \, v + u \, v'\). This tells us that to differentiate the product of two functions \(u\) and \(v\), we need to follow these steps:
Consider the product rule formula: \((u \, v)' = u' \, v + u \, v'\). This tells us that to differentiate the product of two functions \(u\) and \(v\), we need to follow these steps:
- Calculate the derivative of the first function \(u\), which is \(u'\).
- Multiply \(u'\) by the second function \(v\).
- Calculate the derivative of the second function \(v\), which is \(v'\).
- Multiply \(v'\) by the first function \(u\).
- Add these two products together to get the final derivative.
Exploring the Chain Rule
The chain rule is another vital concept in calculus that allows us to differentiate composite functions. A composite function is where one function is inside another, such as \(e^{-x}\) in the example \(y = x e^{-x}\).
To use the chain rule, we follow these steps:
To use the chain rule, we follow these steps:
- Identify the outer function and inner function. For \(e^{-x}\), the outer function is the exponential \(e^x\) and the inner function is \(-x\).
- Differentiate the outer function with respect to the inner function.
- Multiply this derivative by the derivative of the inner function.
Delving into Exponential Functions
Exponential functions are those in which the variable appears as the exponent. A general form of an exponential function is \(f(x) = e^{g(x)}\), making it unique because it grows rapidly.
In our example, \(e^{-x}\) is an exponential function. To differentiate such functions:
In our example, \(e^{-x}\) is an exponential function. To differentiate such functions:
- The derivative of \(e^{g(x)}\) is \((e^{g(x)})' = g'(x) \, e^{g(x)}\).
- The base, \(e\), is a constant approximately equal to 2.718.
- The constant \(e\) ensures that the rate of change of the exponential function is proportional to the value of the function itself.
