91Ó°ÊÓ

An exponential function is a function of the form \(f(x) = a \cdot e^{bx}\), where \(e\) is the base of natural logarithms (approximately 2.71828). In this formula, \(a\) and \(b\) are constants. Exponential functions are unique because their rate of growth or decay is proportional to their current value.

Some key points about exponential functions:

• The base \(e\) ensures that the derivative of the exponential function has a straightforward form.
• Any transformation involving exponential functions, such as shifting or stretching, can be accounted for in these constants.

The exponential function \(e^{3x}\) in our exercise shows how such transformations affect differentiation. By understanding this property, we can easily proceed with the differentiation process.

The chain rule is a fundamental technique in calculus for differentiating composed functions. If you have a function \(g\) composed with another function \(f\) (i.e., \(h(x) = g(f(x))\)), the chain rule states:

\(\frac{d}{dx} g(f(x)) = g'(f(x)) \cdot f'(x) \)

When differentiating \(e^{3x}\), we use the chain rule because \(e^{3x}\) can be seen as an outer function \(e^{u}\) where \(u = 3x\). By the chain rule, the derivative of \(e^{3x}\) is:

\(\frac{d}{dx} e^{3x} = e^{3x} \cdot \frac{d}{dx}(3x) = 3e^{3x} \)

Remember the steps we follow:
1. Differentiate the outer function while keeping the inner function intact.
2. Multiply by the derivative of the inner function.

The constant multiple rule simplifies the differentiation process when a constant is involved. This rule states:

\(\frac{d}{dx} [k \cdot g(x)] = k \cdot \frac{d}{dx} g(x) \)

In other words, if you have a constant \(k\) multiplying a function \(g(x)\), you can take the constant out of the derivative operation, differentiate the function normally, and then multiply the result by the constant.

In our exercise, the function \( f(x) = \frac{1}{3} e^{3x} \) uses the constant multiple rule. First, we note the constant term \(\frac{1}{3}\). Then we differentiate \( e^{3x}\) using the chain rule, as previously described. Lastly, we multiply the derivative of \( e^{3x}\) by \(\frac{1}{3}\):

\( f'(x) = \frac{1}{3} \cdot 3e^{3x} = e^{3x} \)

By combining all these rules, we simplify complex differentiation tasks.