Understanding the Derivative of e^(e^x)

The question asks for the derivative of the function \(y = \rm e^{e^x}\) with respect to \(e^x\). This is different from finding the derivative with respect to \(x\). To solve this, we can use a substitution method.

Applying Substitution for Differentiation

Let's simplify the problem by introducing a new variable.

• Let \(u = e^x\).
• Now, the function \(y = \rm e^{e^x}\) can be written in terms of \(u\) as \(y = e^u\).

We are asked to find the derivative of \(y\) with respect to \(u\), which is denoted as \(\frac{dy}{du}\).

Calculating the Derivative

To find \(\frac{dy}{du}\), we differentiate \(y = e^u\) with respect to \(u\).

The derivative of the exponential function \(e^z\) with respect to \(z\) is \(e^z\).

So, \(\frac{dy}{du} = \frac{d}{du}(e^u) = e^u\).

Substituting Back the Original Variable

Now we substitute back the original expression for \(u\), which is \(u = e^x\).

So, \(\frac{dy}{du} = e^u = e^{e^x}\).

Thus, the derivative of \(\rm e^{e^x}\) with respect to \(\rm e^x\) is \(\rm e^{e^x}\).

Comparing with Options

Let's look at the given options:

1. \( \rm e^{e^x} \)
2. \( \rm e^x \)
3. \( \rm e^{e^x} e^x \)
4. \( \rm ee^x \)

Our calculated derivative, \( \rm e^{e^x} \), matches option 1.

Note that option 3, \( \rm e^{e^x} e^x \), would be the derivative of \( \rm e^{e^x} \) with respect to \(x\), which is found using the chain rule: \(\frac{d}{dx}(e^{e^x}) = e^{e^x} \cdot \frac{d}{dx}(e^x) = e^{e^x} \cdot e^x\).

Therefore, the correct answer is \( \rm e^{e^x} \).

Revision Table: Key Differentiation Rules

Additional Information on Differentiation and Substitution

Differentiation is a fundamental concept in calculus that measures the rate at which a function changes with respect to its input variable. The derivative of a function \(f(x)\) with respect to \(x\) is denoted as \(f'(x)\) or \(\frac{df}{dx}\).

Derivative with Respect to a Function

When asked for the derivative of a function \(y\) with respect to another function of the same variable, say \(u(x)\), you are essentially looking for \(\frac{dy}{du}\). If \(y = f(u)\), then \(\frac{dy}{du} = f'(u)\).

This can be understood using the chain rule. We know that \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). If \(\frac{du}{dx} \neq 0\), we can write \(\frac{dy}{du} = \frac{dy/dx}{du/dx}\). In our case, \(y = e^{e^x}\) and \(u = e^x\). \(\frac{dy}{dx} = e^{e^x} \cdot e^x\) \(\frac{du}{dx} = e^x\) So, \(\frac{dy}{du} = \frac{e^{e^x} \cdot e^x}{e^x} = e^{e^x}\). This confirms the result obtained through direct substitution.

The Exponential Function \(e^x\)

The number \(e\) is a special mathematical constant approximately equal to 2.71828. The function \(f(x) = e^x\) is the unique function that is equal to its own derivative, i.e., \(\frac{d}{dx}(e^x) = e^x\).