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Chapter 4: Problem 22

In Exercises find \(d y / d x\). $$y=e^{1 / x}$$

Short Answer

Expert verified
\( \frac{dy}{dx} = -\frac{e^{1/x}}{x^2} \).

Step by step solution

01

Identify the Function and Inner Function

The given function is \( y = e^{1/x} \). Here, \( 1/x \) is the inner function. We see that \( y \) is an exponential function with the form \( e^u \), where \( u = 1/x \).
02

Differentiate the Outer Function

First, differentiate the outer function \( e^u \) with respect to \( u \). The derivative of \( e^u \) is \( e^u \).
03

Differentiate the Inner Function

Now, differentiate the inner function \( u = 1/x \) with respect to \( x \). The derivative of \( 1/x \) is \( -1/x^2 \).
04

Apply the Chain Rule

Apply the chain rule for differentiation, which states that if \( y = e^{u} \) and \( u = 1/x \), then \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). Substituting the values, we have \( \frac{dy}{dx} = e^{1/x} \cdot \left(-\frac{1}{x^2}\right) \).
05

Simplify the Expression

Finally, simplify the expression to find \( \frac{dy}{dx} = -\frac{e^{1/x}}{x^2} \).

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Key Concepts

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

Exponential Function
An exponential function is a type of mathematical function where a constant base is raised to a variable exponent. In the case of the given exercise, our exponential function is expressed as \( e^u \), where \( e \) is the natural exponential base and \( u \) is any real number or expression. The base \( e \) is an important constant approximately equal to 2.718, and it appears frequently in various areas of mathematics.
  • Exponential functions grow at a rate proportional to their current value. This unique property makes them applicable in natural growth processes, such as population growth or radioactive decay.
  • Derivatives of exponential functions maintain a similar form. For any function of the type \( e^u \), where \( u \) is a function of \( x \), the derivative with respect to \( u \) is still \( e^u \).
These properties make understanding exponential functions essential for solving problems involving exponential growth or decay, as well as for complex calculus and differential equations.
Chain Rule
The chain rule is a fundamental concept in calculus used for differentiating compositions of functions. When you have a function within another function 鈥� known as a composite function 鈥� the chain rule helps us find the derivative effectively. It essentially tells us how to differentiate a function of a function.
  • The chain rule formula is stated as \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \), where \( y \) is the outer function and \( u \) is the inner function.
  • This rule is particularly useful when dealing with complicated derivatives, to break them down into simpler parts.
  • In the exercise provided, the chain rule allows us to differentiate the expression \( y = e^{1/x} \) by treating \( 1/x \) as \( u \), differentiating each part separately.
Mastering the chain rule is essential for tackling more complex calculus problems effectively.
Inner Function
The inner function is the function that is embedded within another function in composite functions. In the provided exercise, the inner function is \( u = \frac{1}{x} \). Identifying the inner function is a crucial step in using the chain rule for differentiation.
  • An inner function is differentiated with respect to its own variable, which in this case, results in the derivative \( \frac{du}{dx} = -\frac{1}{x^2} \).
  • This step is aligned with the process of finding the derivative of a composite function by breaking down the functions into manageable parts.
Understanding the role of the inner function makes it easier to apply the chain rule and arrive at the correct derivative when dealing with functions enclosed within others.
Outer Function
The outer function in a composite function is the function that envelops the inner function. For the expression \( y = e^{1/x} \), the outer function is \( y = e^u \) after substituting \( u = 1/x \). Differentiating the outer function is usually the first step after identifying the inner function.
  • The derivative of the exponential outer function \( e^u \) with respect to \( u \) is simply \( e^u \), demonstrating the self-similar property of exponential functions.
  • After differentiating the outer function, this result is used in the chain rule to combine with the derivative of the inner function.
Clearly understanding the outer function's role allows one to apply the chain rule systematically, ensuring accurate and efficient differentiation of composite expressions.

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