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Chapter 3: Problem 100

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

Short Answer

Expert verified
The derivative of the function is: \[ f'(x) = e^{e^{x} + x} \]

Step by step solution

01

Identify the function

The given function is: \[ f(x) = e^{e^{x}} \]
02

Apply the chain rule

To differentiate \( f(x) = e^{e^{x}} \), use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then \( f'(x) = f'(g(x)) \times g'(x) \).
03

Differentiate the outer function

First, differentiate the outer function. Let \( u = e^{x} \). Then, the function becomes \( e^u \). The derivative of \( e^u \) with respect to \( u \) is \( e^u \).
04

Differentiate the inner function

Next, differentiate the inner function. The inner function is \( e^{x} \). The derivative of \( e^{x} \) with respect to \( x \) is \( e^{x} \).
05

Combine the derivatives

Using the chain rule, combine the two derivatives: \[ f'(x) = (e^{e^{x}}) \times (e^{x}) \]
06

Simplify the expression

Therefore, the derivative of the function \( f(x) = e^{e^{x}} \) is: \[ f'(x) = e^{e^{x} + x} \]

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Key Concepts

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

The Chain Rule
To differentiate complex functions, the chain rule is essential. It's especially useful when dealing with composite functions. The chain rule states that if you have a function composed of two functions, say \( f(g(x)) \), then the derivative is given by \( f'(x) = f'(g(x)) \times g'(x) \). This rule helps you break down the task into manageable pieces. Understanding this concept is key to tackling functions within functions. By focusing on the outer and inner functions separately, you can simplify the differentiation process.
Composite Functions
A composite function involves a function within another function, like \( f(g(x)) \). In our example, \( f(x) = e^{e^{x}} \), \( g(x) = e^{x} \) is inside the exponential function. To differentiate, treat each function step by step. First, treat the outer function \( e^u \) where \( u = e^{x} \), and then handle the inner function \( e^{x} \). This layered approach makes the problem easier to solve. Understanding composite functions lets you take complex operations apart, making differentiation straightforward.
Exponential Functions
Exponential functions have the form \( e^x \), where \( e \) is the base of the natural logarithm. The great thing about these functions is their derivative is the same as the original function: \( \frac{d}{dx} e^x = e^x \). This is what makes exponential functions unique and easier to handle in differentiation. In our problem, \( e^{e^{x}} \), the exponential functions stack up. But by differentiating them using the chain rule, you can manage them efficiently. Exponential functions often appear in various fields, so mastering their differentiation is very useful.
Derivatives
Derivatives represent the rate at which a function changes. They are crucial in calculus for understanding how functions behave. In our example, differentiating \( f(x) = e^{e^{x}} \) lets us see how this function evolves as \( x \) changes. Following the step-by-step method, we get the derivative: \( f'(x) = e^{e^{x} + x} \). This expression tells us the instant rate of change for \( f(x) \) at any point \( x \). Understanding derivatives is fundamental for solving practical problems involving motion, growth, and more.

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Most popular questions from this chapter

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