What is the derivative of eˣ with respect to eˣ?

• A: x eˣ / eˣ
• B: eˣ / x eˣ
• C: x eˣ / x eˣ
• D: eˣ / eˣ

The derivative of a function measures how that function changes as its input changes. In this case, we want to find the derivative of \( e^x \) with respect to \( e^x \). To do this, we can apply the chain rule of differentiation.

Understanding the Derivative

The function \( e^x \) is unique because its derivative is itself. Therefore, when we differentiate \( e^x \) with respect to \( e^x \), we can express it in terms of the variable \( e^x \).

Applying the Chain Rule

Using the chain rule, we can say:

• The derivative of \( e^x \) is \( e^x \).
• We are differentiating with respect to \( e^x \), which means we treat \( e^x \) as the variable.

Final Calculation

Thus, the derivative of \( e^x \) with respect to \( e^x \) simplifies to:

1

Choosing the Correct Option

Looking at the options provided:

• A: \( \frac{x e^x}{e^x} \)
• B: \( \frac{e^x}{x e^x} \)
• C: \( \frac{x e^x}{x e^x} \)
• D: \( \frac{e^x}{e^x} \)

The correct answer is D: \( \frac{e^x}{e^x} \), which simplifies to 1.