Problem 21 Differentiate the following func... [FREE SOLUTION]

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Calculus and Its Applications

Larry J. Goldstein, David I. Schneider, Nakhle H. Asmar

$Math Studyset 91影视 Explanations$ Math

13 Edition

Chapter 4: Problem 21

Differentiate the following functions. $f(x)=e^{e^{2}}$

Short Answer

Expert verified

The derivative of the function is 0.

Step by step solution

Identify the Function Components

The given function is a constant function because the exponent does not vary with respect to x.

Differentiate the Constant Function

Since the exponent of the function is a constant value, the derivative of a constant is always 0.

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Key Concepts

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

Constant Function

A constant function is a function that always returns the same value, no matter what the input is. For example, if we have a function defined as $f(x) = c$ where $c$ is a constant, the output of $f(x)$ does not change with different values of $x$. This means the graph of a constant function is a horizontal line.

For our exercise, the function given is $f(x) = e^{e^2}$. Here, $e^{e^2}$ is a constant because it does not vary with $x$. It's just a number, even though it looks complicated.

Example: $f(x) = 5$
No matter what $x$ is, $f(x)$ is always 5.

Exponential Function

Exponential functions have the form $f(x)=a^{x}$ where $a$ is a positive constant. They show rapid growth or decline as $x$ changes.

In exponential functions, the base $a$ is raised to the power of $x$. However, our given function has the base $e$ (Euler's number) raised to another constant $e^2$.

Although it seems similar, $e^{e^2}$ is not a typical exponential function since $x$ is not part of the exponent. Thus, it's treated as a constant rather than an exponential function that depends on $x$.

Derivative

The derivative of a function gives us the rate at which the function's value changes with respect to changes in its input. For a function $f(x)$, the derivative is denoted as $f'(x)$ or $\frac{df(x)}{dx}$.

When differentiating constant functions like $f(x) = c$, the derivative is always zero because a constant does not change, hence its rate of change is zero.

In our case: $f(x) = e^{e^2}$. Since $e^{e^2}$ is a constant, $f'(x) = 0$.

Examples: If $h(x) = 7$, $h'(x) = 0$
If $g(x) = \frac{1}{2}$, $g'(x) = 0$

The rule here is straightforward: the derivative of any constant is zero.

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91影视

Differentiate the following functions. \(f(x)=e^{e^{2}}\)

Short Answer

Step by step solution

Identify the Function Components

Differentiate the Constant Function

Key Concepts

Constant Function

Exponential Function

Derivative

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