What is the derivative of e^-1?

The derivative of a constant is always zero, and since \( e^{-1} \) is a constant value, its derivative is also zero. To break this down further, let's look at what \( e^{-1} \) represents and how derivatives work in general.

Understanding Constants and Derivatives

In calculus, the derivative measures how a function changes as its input changes. When we talk about constants, like \( e^{-1} \), we are referring to fixed values that do not change regardless of the input. The number \( e \) is approximately 2.71828, and \( e^{-1} \) is simply \( \frac{1}{e} \), which is approximately 0.367879.

Why is the Derivative of a Constant Zero?

To grasp why the derivative of a constant is zero, consider the definition of a derivative:

• The derivative of a function \( f(x) \) at a point \( x \) is defined as the limit of the average rate of change of the function as the interval approaches zero.
• Mathematically, this is expressed as: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h].

When \( f(x) \) is a constant, say \( c \), then \( f(x + h) = c \) and \( f(x) = c \). Thus, the expression simplifies to:

f'(x) = lim (h -> 0) [(c - c) / h] = lim (h -> 0) [0 / h] = 0.

Applying This to \( e^{-1} \)

Since \( e^{-1} \) is a constant, we can apply the same reasoning. The derivative of \( e^{-1} \) is:

f'(x) = 0.

Visualizing the Concept

Imagine a flat horizontal line on a graph representing the constant value \( e^{-1} \). No matter how far you move along the x-axis, the height of the line remains the same. This illustrates that there is no change in the value, hence the slope (derivative) is zero.

Summary

In summary, the derivative of \( e^{-1} \) is zero because it is a constant value. Understanding this concept is fundamental in calculus, as it lays the groundwork for more complex functions and their derivatives. If you have any more questions about derivatives or any other calculus topics, feel free to ask!