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Magazine Chapter 3: Problem 70
Find the thousandth derivative of \(f(x)=x e^{-x}\)
Short Answer
Expert verified
The thousandth derivative is \(e^{-x}(x-1000)\).
Step by step solution
01
Identify the function type
The given function is in the form of a product: \(f(x) = x e^{-x}\)We note it has both a polynomial factor \(x\) and an exponential factor \(e^{-x}\).
02
Apply the Product Rule for the First Derivative
Using the product rule, the derivative of \(f(x) = x e^{-x}\) is computed as:\(f'(x) = \frac{d}{dx}(x) * e^{-x} + x * \frac{d}{dx}(e^{-x})\) which simplifies to:\(f'(x) = e^{-x} - x e^{-x}\).
03
Recursively Apply the Derivative
Notice how each derivative will follow a recursive pattern. For the second derivative:\(f''(x) = - e^{-x} + (1 + x) e^{-x}\), simplifying to:\(f''(x) = x e^{-x} - 2e^{-x}\).Each derivative reduces the power of \(x\) by one until reaching 0.
04
Identify the Pattern
Each time a derivative is taken, the result follows:\(f^{(n)}(x) = (-1)^n e^{-x}(x-n)\).Thus, the original polynomial factor decreases by 1 with each derivative taken.
05
Calculate the Thousandth Derivative
Using the pattern derived, the thousandth derivative is:\(f^{(1000)}(x) = (-1)^{1000} e^{-x}(x-1000)\)Since \((-1)^{1000} = 1\), we have:\(f^{(1000)}(x) = e^{-x}(x-1000)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative
Derivatives are a fundamental concept in calculus. They represent the rate at which a function is changing at any given point. For a function of a single variable, the derivative is the slope of the tangent line to the function's graph at a particular point.
In practical terms, if you have a function \(f(x)\), its derivative, often denoted as \(f'(x)\), can tell you how \(f\) is going to behave with small changes in \(x\).
In practical terms, if you have a function \(f(x)\), its derivative, often denoted as \(f'(x)\), can tell you how \(f\) is going to behave with small changes in \(x\).
- Simple derivatives like the derivative of \(x^2\) lead to expressions like \(2x\).
- Derivatives can predict the maxima and minima of functions.
- They also appear in optimization problems where something needs to be maximized or minimized.
Product Rule
The product rule is a technique used to find the derivative of products of two functions. In our current exercise, \(f(x) = x e^{-x}\), the product rule becomes essential because the function involves the multiplication of two parts: a polynomial \(x\) and an exponential \(e^{-x}\).
The product rule is usually expressed as:\[ (uv)' = u'v + uv' \]Here \(u\) and \(v\) are functions of \(x\). Their derivatives are \(u'\) and \(v'\), respectively.
The product rule is usually expressed as:\[ (uv)' = u'v + uv' \]Here \(u\) and \(v\) are functions of \(x\). Their derivatives are \(u'\) and \(v'\), respectively.
- Apply it to our function, we differentiate \(x\) to get \(1\), and \(e^{-x}\) to get \(-e^{-x}\).
- The rule ensures we differentiate each component independently and then combine them appropriately.
- By using it, the initial derivative turns into \(e^{-x} - x e^{-x}\).
Pattern Recognition
Pattern recognition involves identifying regularities or common themes that simplify the process of calculating derivatives, especially in repetitive derivative tasks.In the given problem, after computing several derivatives, a pattern emerges.
Examine how each time a derivative is taken, the power of \(x\) decreases, and a factor involving \((-1)\) appears, capturing the alternating nature of signs in successive derivatives.
Examine how each time a derivative is taken, the power of \(x\) decreases, and a factor involving \((-1)\) appears, capturing the alternating nature of signs in successive derivatives.
- Recognizing such patterns is powerful, as it transforms a potentially tedious process into a straightforward calculation.
- It reveals the formula for the \(n\)-th derivative as \((-1)^n e^{-x}(x-n)\), which guides us to easily find higher order derivatives.
- This pattern streamlines the process, making it possible to directly calculate the thousandth derivative without computing all previous ones.
Exponential Function
Exponential functions are functions given by expressions like \(a^x\) or \(e^x\). Here, \(e\) represents Euler's number, approximately equal to \(2.718\), which is the base of the natural logarithm.
In calculus, exponential functions are prominent due to their unique property: the rate of change of \(e^x\) is proportional to its current value, meaning \(\frac{d}{dx} e^x = e^x\).
In calculus, exponential functions are prominent due to their unique property: the rate of change of \(e^x\) is proportional to its current value, meaning \(\frac{d}{dx} e^x = e^x\).
- This characteristic makes them well-suited for modeling growth and decay processes, like population growth or radioactive decay.
- When combined with other functions, like \(x e^{-x}\), understanding their derivative behavior requires integrating them with rules such as the product rule.
- In our exercise, the term \(e^{-x}\) introduces a decreasing exponential component, affecting each derivative calculation.
- It remains consistent as a multiplying factor in each derivative due to its derivative being a simple scaling transformation.
