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Chapter 3: Problem 44

If \(g(x)=x / e^{x}, \text { find } g^{(n)}(x)\)

Short Answer

Expert verified
\( g^{(n)}(x) = \frac{n-x}{e^x} \) for any integer \( n \).

Step by step solution

01

Understand the Function

The function provided is \( g(x) = \frac{x}{e^x} \). This means we need to calculate the derivatives of this function multiple times to find a pattern for \( g^{(n)}(x) \).
02

Calculate the First Derivative

To find \( g'(x) \), we use the quotient rule, which is \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \). Here, \( u = x \) and \( v = e^x \). Thus, \( u' = 1 \) and \( v' = e^x \). The first derivative is:\[g'(x) = \frac{1 \cdot e^x - x \cdot e^x}{(e^x)^2} = \frac{e^x - xe^x}{e^{2x}} = \frac{e^x(1 - x)}{e^{2x}} = \frac{1 - x}{e^x}\]
03

Calculate the Second Derivative

Apply the quotient rule again to \( g'(x) = \frac{1-x}{e^x} \) with \( u = 1-x \) and \( v = e^x \). Thus, \( u' = -1 \) and \( v' = e^x \). The second derivative is:\[g''(x) = \frac{-1 \cdot e^x - (1-x)\cdot e^x}{(e^x)^2} = \frac{-e^x - e^x + xe^x}{e^{2x}} = \frac{x-2}{e^x}\]
04

Calculate the Third Derivative

Apply the quotient rule again to \( g''(x) = \frac{x-2}{e^x} \) with \( u = x-2 \) and \( v = e^x \). Thus, \( u' = 1 \) and \( v' = e^x \). The third derivative is:\[g'''(x) = \frac{1 \cdot e^x - (x-2)\cdot e^x}{(e^x)^2} = \frac{e^x - xe^x + 2e^x}{e^{2x}} = \frac{3-x}{e^x}\]
05

Identify and Confirm the Pattern

The pattern appears to be \( g^{(n)}(x) = \frac{n-x}{e^x} \). To confirm, it is helpful to derive \( g^{(4)}(x) \) using the same method:\[g^{(4)}(x) = \frac{4-x}{e^x}\]Thus, \( g^{(n)}(x) \) consistently fits this pattern for successive derivatives.
06

General Solution

The nth derivative of the function \( g(x) \) is \( g^{(n)}(x) = \frac{n-x}{e^x} \). This pattern matches for derivatives of all successive orders and confirms the relationship observed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule
The quotient rule is a fundamental tool in calculus, especially when dealing with functions expressed as the division of two sub-functions. When you have a function in the form \( g(x) = \frac{u}{v} \), where both \( u \) and \( v \) are differentiable functions of \( x \), you apply the quotient rule to find its derivative. The rule states:
  • First, differentiate the numerator, \( u' \).
  • Then, differentiate the denominator, \( v' \).
  • The quotient rule formula is: \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \).
In our example, the quotient rule is repeatedly applied, simplifying the process of finding higher-order derivatives. The simplicity and repetitive nature of the rule make it a reliable method when working with complex fractions.
Pattern Recognition in Derivatives
Identifying patterns in derivatives is an invaluable skill, particularly when dealing with higher-order derivatives. After computing the first few derivatives, you may start noticing a regular structure or behavior in the outcomes. This occurs because of consistent applications of differentiation rules, like the quotient rule.
  • Start with identifying the base function and its derivatives.
  • Look for repeating components or constants in the derivative expressions.
  • In our function, we noticed the recurring representation \( g^{(n)}(x) = \frac{n-x}{e^x} \).
By systematically calculating derivatives and observing such patterns, you can significantly speed up the process of finding high-order derivatives without recalculating each one individually.
Calculus for Life Sciences
In life sciences, calculus is used to model and analyze biological systems, involving rates of change and accumulation. Functions like \( g(x) = \frac{x}{e^x} \) might describe biological phenomena such as growth rates or decay processes. The capability to differentiate such functions helps us to understand key behaviors and progressions over time.
  • Understand how quantities change in time or space.
  • Predict the impact of changes in biological systems.
  • Apply calculus to areas like population dynamics or pharmacokinetics.
By mastering derivatives and recognizing patterns, students in the life sciences can make quantitative predictions and contribute to fields like biology and medicine.
Function Differentiation
Function differentiation is at the core of calculus, allowing us to understand how a function behaves as its input changes. To differentiate a function means to determine its rate of change, providing insights into its slope and curvature.
  • Begin with simple functions and progress to more complex forms.
  • Utilize rules like the quotient, product, and chain to find derivatives efficiently.
  • Evaluate the behavior of the function across its domain through derivatives.
In our solved exercise, function differentiation led us through successive derivatives to observe a general pattern. This approach not only reveals the essence of the function but also prepares students for advanced topics in calculus.

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Most popular questions from this chapter

Gene regulation Genes produce molecules called mRNA that go on to produce proteins. High concentrations of protein inhibit the production of mRNA, leading to stable gene regulation. This process has been modeled (see Section 10.3 ) to show that the concentration of mRNA overtime is given by the equation \(m(t)=\frac{1}{2} e^{-t}(\sin t-\cos t)+\frac{1}{2}\) What is the rate of change of mRNA concentration as a function of time?

Show that the curve \(y=6 x^{3}+5 x-3\) has no tangent line with slope \(4 .\)

(a) Use the definition of the derivative to calculate \(f^{\prime}\) , (b) Check to see that your answer is reasonable by comparing the graphs of \(f\) and \(f^{\prime}\) \(f(t)=t^{2}-\sqrt{t}\)

Suppose \(f\) is differentiable on \(\mathbb{R} .\) Let \(F(x)=f\left(e^{x}\right)\) and \(G(x)=e^{f(x)} .\) Find expressions for \((a) F^{\prime}(x)\) and \((b) G^{\prime}(x)\).

Under certain circumstances a rumor spreads according to the equation \(p(t)=\frac{1}{1+a e^{-k t}}\) where \(p(t)\) is the proportion of the population that has heard the rumor at time \(t\) and \(a\) and \(k\) are positive constants. (a)find lim \(_{t \rightarrow \infty} p(t)\) (b)Find the rate of spread of the rumor. (c)Graph \(p\) for the case \(a=10, k=0.5\) with \(t\) measured in hours. Use the graph to estimate how long it will take for 80\(\%\) of the population to hear the rumor.
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