91Ó°ÊÓ

Logarithmic differentiation is a helpful technique to differentiate complex functions involving products, quotients, or powers. When dealing with a function that is difficult to differentiate directly, we can use logarithms to simplify the expression. This method leverages the properties of logarithms such as \(\text{log}_{a}(xy) = \text{log}_{a}(x) + \text{log}_{a}(y)\) and \(\text{log}_{a}(\frac{x}{y}) = \text{log}_{a}(x) - \text{log}_{a}(y)\). By taking the natural logarithm of both sides of the equation, we can use the properties of logarithms to make differentiation easier.
In our exercise, we start with the function \(y = \text{ln}(\frac{x}{2})\). Using the property of logarithms for a fraction, we rewrite this as \(y = \text{ln}(x) - \text{ln}(2)\). This simplification allows us to differentiate each term separately, making use of the familiar differentiation rules.
We'll introduce differentiation rules next.

Differentiation rules are essential tools for finding the derivative of a function. Some of the most important rules include:

• Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). This rule applies to any real number \(n\).
• Product Rule: If \(f(x) = u(x) \times v(x)\), then \(f'(x) = u'(x)v(x) + u(x)v'(x)\).
• Quotient Rule: If \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).
• Chain Rule: If \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x)) \times h'(x)\).
• Logarithmic Differentiation: As discussed, taking the logarithm of both sides of an equation allows the application of simpler rules to complex functions.

Applying these rules allows us to tackle a wide variety of functions. In our exercise, we specifically use the properties of logarithms to simplify before applying the power rule.
We'll look deeper into the power rule in the next section.

The power rule is one of the simplest and most commonly used differentiation rules. It states that if you have a function of the form \(f(x) = x^n\), the derivative is found by multiplying the power \(n\) by the function and then reducing the power by one:
\[ \frac{d}{dx}(x^n) = nx^{n-1} \]
This rule is easy to apply and is powerful for differentiating polynomial functions. In our exercise, after simplifying the logarithmic function, we end up with \( y = \text{ln}(x) - \text{ln}(2)\). The derivative of \(\text{ln}(x)\) is \( \frac{1}{x} \), and since \( \text{ln}(2) \) is a constant, its derivative is 0:
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