91Ó°ÊÓ

Exponential functions are a fundamental part of calculus and appear frequently in many areas of mathematics and science. An exponential function is typically written in the form: \( f(x) = e^x \) where \( e \) is Euler's number, approximately equal to 2.71828. The key characteristic of exponential functions is that their rate of change is proportional to their value.

In this context, \( e^x \) means that the function grows (or decays) at a rate proportional to its current value, which is unique and powerful in modeling many real-world phenomena such as population growth, radioactive decay, and interest calculations.

The constant multiple rule is a simple yet powerful tool in differentiation. It states that when you have a constant multiplied by a function, the derivative of this product is the constant multiplied by the derivative of the function. Mathematically, this is expressed as:

\[\frac{d}{dx}[a \times f(x)] = a \times f'(x)\]

For example, if we need to differentiate \( 6e^x \), we can take the constant 6 outside the differentiation process, and then differentiate \( e^x \) as usual. This rule simplifies many differentiation problems, making it easier to find the resulting derivative without complicated calculations.

Calculating the derivative of a function is one of the core tasks in calculus. Let’s walk through the derivative calculation of the given function \( f(x) = 6e^x \).

1. **Identify the function**: We start with the function \( f(x) = 6e^x \).
2. **Differentiate \( e^x \)**: The derivative of the exponential function \( e^x \) with respect to \( x \) is simply \( e^x \).
3. **Apply the constant multiple rule**: Since our function involves a constant multiple (6), we use the constant multiple rule:
\[ \frac{d}{dx}[6 e^x] = 6 \times \frac{d}{dx}[e^x] \]
4. **Combine the results**: We know \( \frac{d}{dx}[e^x] = e^x \). Therefore:
\[ \frac{d}{dx}[6 e^x] = 6 e^x \]
In conclusion, the derivative of \( f(x) = 6e^x \) is \( f'(x) = 6e^x \).