91Ó°ÊÓ

The natural logarithm, denoted as \text{ln}\, is the logarithm to the base \(e\), where \(e\approx2.71828\). This function is widely used in calculus. One key property of natural logarithms is their ability to simplify the differentiation of complex functions.
When using logarithmic differentiation, the function simplifies due to the properties of logarithms, making it easier to handle exponential terms. For example, in the equation \(f(x) = 2^x\), taking the natural logarithm of both sides results in \(\text{ln}(f(x)) = \text{ln}(2^x)\). This simplification is crucial for subsequent steps.

The chain rule is a fundamental technique in differentiation, especially when dealing with composite functions. It states that if a function \(y\) depends on \(u\), which in turn depends on \(x\), then the derivative of \(y\) with respect to \(x\) is the product of the derivative of \(y\) with respect to \(u\) and the derivative of \(u\) with respect to \(x\):
\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]
In the given solution, the chain rule is used when differentiating \(\text{ln}(f(x))\). First, we differentiate the outer function \(\text{ln}(u)\) with respect to \(u\), yielding \(1/f(x)\). Then, we multiply by the derivative of the inner function \(f(x)\) with respect to \(x\) (\(f'(x)\)). Thus, we get:
\[\frac{d}{dx} \big( \text{ln}(f(x)) \big) = \frac{f'(x)}{f(x)}\]

Logarithm properties play a crucial role in simplifying expressions for easier differentiation. The primary properties used in the solution are:
• Power Rule: \(\text{ln}(a^b) = b \text{ln}(a)\)
Applying this property to our function gives us:\[\text{ln}(2^x) = x \cdot \text{ln}(2)\]which simplifies the differentiation process.

Understanding these properties allows us to rewrite the expression in a more manageable form. This is an essential step in logarithmic differentiation, making subsequent calculations straightforward.