Problem 70 - Essential Calculus Early Transcendentals

Question

Find the thousandth derivative of \(f(x)=x e^{-x}\)

Step-by-Step Solution

Answer

The thousandth derivative is \(e^{-x}(x-1000)\).

Step 1: 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 \(e^{-x}\).

Step 2: 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:\(f'(x) = e^{-x} - x e^{-x}\).

Step 3: 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.

Step 4: 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.

Step 5: 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:\(f^{(1000)}(x) = e^{-x}(x-1000)\).

Key Concepts

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 \(x\).

• 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.

For our exercise, finding the derivative of a product involves a specific technique called the product rule, which is crucial for dealing with functions like \(f(x) = x e^{-x}\).

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.

• 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 \(e^{-x} - x e^{-x}\).

This shows how the product rule simplifies finding derivatives of complex functions.

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.

• 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 \(\frac{d}{dx} e^x = e^x\).

• 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.

Exponential functions thus play a crucial role in shaping the behavior of derivatives in calculus problems.