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Chapter 4: Problem 28

Differentiate. $$ f(x)=\frac{e^{x}}{x^{5}} $$

Short Answer

Expert verified
\[ f'(x) = \frac{e^x (x - 5)}{x^6} \]

Step by step solution

01

- Identify the Function

Given the function to differentiate: \[ f(x) = \frac{e^x}{x^5} \] This is a quotient of two functions: the numerator is \( e^x \) and the denominator is \( x^5 \).
02

- Use the Quotient Rule

The Quotient Rule for differentiation states: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \] Here, \( u = e^x \) and \( v = x^5 \).
03

- Differentiate the Numerator and the Denominator

Differentiate \( u = e^x \): \[ u' = e^x \] Differentiate \( v = x^5 \): \[ v' = 5x^4 \]
04

- Apply the Quotient Rule

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula: \[ f'(x) = \frac{e^x \cdot x^5 - e^x \cdot 5x^4}{x^{10}} \]
05

- Simplify the Expression

Factor out \( e^x \) in the numerator: \[ f'(x) = \frac{e^x (x^5 - 5x^4)}{x^{10}} \] Simplify inside the parentheses: \[ x^5 - 5x^4 = x^4(x - 5) \] Thus, \[ f'(x) = \frac{e^x \cdot x^4 (x - 5)}{x^{10}} \]
06

- Final Simplification

Split the fraction: \[ f'(x) = e^x \cdot \frac{x^4 (x - 5)}{x^{10}} \] Cancel out \( x^4 \) in the numerator and denominator: \[ f'(x) = e^x \cdot \frac{(x - 5)}{x^6} \] Therefore, \[ f'(x) = \frac{e^x (x - 5)}{x^6} \]

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Key Concepts

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

Quotient Rule
When we need to differentiate a function that is a quotient, such as \( \frac{u}{v} \), we use the Quotient Rule. The rule is crucial for working with functions where one expression is divided by another. The formula is: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
Here's a breakdown to understand its components:
  • \textbf{u}: The numerator function
  • \textbf{v}: The denominator function
  • \textbf{u'}: The derivative of the numerator function
  • \textbf{v'}: The derivative of the denominator function
Using this rule, we can differentiate complex functions step-by-step. This ensures we correctly account for how each part of the function changes.
Exponential Functions
Exponential functions, like \ e^x \ in this example, involve the constant \ e \ (approximately 2.71828). These functions grow rapidly and have significant mathematical properties. The key derivative to remember is: \[ \frac{d}{dx} e^x = e^x \]
When differentiating exponential functions within compound expressions (like our quotient), keep their unique behavior in mind: they retain their form even after differentiation. This characteristic can simplify many calculus problems, making exponential functions intriguing and powerful.
Derivatives
A derivative tells us the rate at which a function is changing at any given point. When taking derivatives, we apply specific rules, such as the power rule, product rule, and quotient rule. In the context of the given problem:
  • Derive \( u = e^x \): \ u' = e^x \.
  • Derive \( v = x^5 \): \ v' = 5x^4 \.
With derivatives of both parts, we apply the Quotient Rule to find the overall derivative. Remembering these fundamental rules enables us to tackle various functions systematically and accurately.

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Most popular questions from this chapter

Differentiate. $$ y=1-e^{-3 x} $$

The growth of maize (corn) leaves is modeled by the logistic equation. For the sixth leaf on a maize plant, it was found that the function $$ A(t)=\frac{105}{1+32900 e^{-0.04 t}} $$ is a good approximation of the leaf's area \(A(t)\) (in \(\mathrm{cm}^{2}\) ) from the time the leal forms until it stops growing. \({ }^{23}\) The number \(t\) is the number of degree days since the plant sprouted. (Degree days measure time weighted by the temperature. The number of degree days for maize is computed by multiplying the time by the number of degrees above \(8^{\circ} \mathrm{C}\).) a) Use the logistic equation to approximate the leaf area after 100 degree days, after 150 degree days, and after 200 degree days. b) Find the rate of change of the area \(A^{\prime}(t)\). c) Find the point of inflection for the function \(A(t)\) d) Sketch the graph of \(A(t)\). e) According to this model, what is the maximum area of the sixth leal on a maize plant?

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