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Logarithmic differentiation is a powerful technique in calculus used to differentiate functions with logarithms. It's particularly useful when dealing with products, quotients, or powers of functions. The basic idea is to take the natural logarithm of both sides of an equation and then differentiate implicitly. Here’s how you can apply this method to the given function:
Consider the function given: \( f(x) = 2 \, \log x \).
Notice that this is a simple logarithmic function multiplied by a constant. To differentiate this using logarithmic differentiation, follow these steps:

• Identify the logarithmic function and the constant.
• Recall the derivative of \( \log x \), which is \( \frac{1}{x} \).
• Apply the constant multiple rule for simplicity.

In this case, using logarithmic differentiation directly might not be necessary, since the function is straightforward. However, understanding this technique is crucial for more complex functions involving logarithms.

The constant multiple rule in calculus states that the derivative of a constant times a function is the constant times the derivative of the function. This rule simplifies differentiation when a function includes a constant factor. Here’s how it works:

Given a function of the form \( f(x) = c \, g(x) \), where \( c \) is a constant and \( g(x) \) is a function of \( x \), the derivative is:
\[ \frac{d}{dx} [c \, g(x)] = c \, \frac{d}{dx} [g(x)] \]
In the given problem, we have \( f(x) = 2 \, \log x \). Apply the constant multiple rule:

• The constant \( c = 2 \).
• The function \( g(x) = \log x \).
• The derivative of \( \log x \) is \( \frac{1}{x} \).

So, the derivative becomes:
\[ \frac{d}{dx} [2 \, \log x] = 2 \, \frac{d}{dx} [\log x] = 2 \, \frac{1}{x} \]
Which simplifies to \( \frac{2}{x} \).

Calculus is the branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The main techniques used in calculus for solving problems are differentiation and integration. In this particular problem, we are focused on differentiation. Differentiation helps in understanding how a function changes at any given point. Here are some fundamental concepts:

• **Derivative**: A measure of how a function changes as its input changes. Denoted as \( f'(x) \) or \( \frac{d}{dx} [f(x)] \).
• **Derivative of a Logarithmic Function**: For \( \log x \), the derivative is \( \frac{1}{x} \).
• **Constant Multiple Rule**: The derivative of a constant times a function is the constant times the derivative of the function.

Using these principles, we can solve complex problems by breaking them down into simpler parts, as shown in the solution to the given exercise. Mastering these basics will help you tackle more challenging calculus problems efficiently.