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Understanding the derivative of a function is crucial in calculus. Let's start with the basics. A derivative represents the rate at which a function changes. It tells you how the value of the function is changing at any given point. For example, if you have a function that measures distance over time, its derivative would represent the speed.

Derivatives have many applications, including finding the slope of a curve, determining velocity, and optimizing functions.

In our specific example, the function is given by \( f(x) = x \, \text{ln}(x) \). To find its derivative, we use the **Product Rule in calculus**, which is applied when differentiating a product of two functions. The product rule formula is: \[ (uv)' = u'v + uv' \].

For \( f(x) = x \, \text{ln}(x) \), we treat \( u = x \) and \( v = \text{ln}(x) \). The derivative of \( x \) (which is \( u' \)) is 1, and the derivative of \( \text{ln}(x) \) (which is \( v' \)) is \( \frac{1}{x} \). Using the product rule, the derivative \( f'(x) = 1 \cdot \text{ln}(x) + x \cdot \frac{1}{x} = \text{ln}(x) + 1 \).

Understanding how to find the derivative is essential for graphing and analyzing the function’s behavior.

Graphing functions is a helpful way to visualize how they behave. When you graph a function, you plot its values over a range of inputs. This helps you see trends, peaks, and valleys. To graph a function like \( f(x) = x \, \text{ln}(x) \), follow these steps:

• Open a graphing calculator or software.
• Enter the function \( f(x) = x \, \text{ln}(x) \).
• Choose an appropriate range for \( x \), excluding 0 since \( \text{ln}(x) \) is not defined at \( x = 0 \). A typical range could be from 0.1 to 10.

The graph of \( f(x) = x \, \text{ln}(x) \) will increase and then decrease after a certain point. This is because \( \text{ln}(x) \) grows very slowly, and at some point, the multiplying factor \( x \) starts to decrease the value.

Next, graph the derivative function \( f'(x) = \text{ln}(x) + 1 \). This graph should be an increasing function crossing the horizontal axis at \( x = \frac{1}{e} \).
Graphing both the function and its derivative together helps to understand how the function’s rate of change behaves across different values of \( x \).

The natural logarithm, denoted as \( \text{ln}(x) \), is a fundamental function in calculus. It is the inverse of the exponential function \( e^x \). In other words, \( \text{ln}(e^x) = x \) and \( e^{\text{ln}(x)} = x \).

Here are some key properties of \( \text{ln}(x) \):

• **Domain and Range:** The function \( \text{ln}(x) \) is defined for all positive values of \( x \). Its domain is \( (0, \infty) \) and its range is \( (-\infty, \infty) \).
• **Derivative:** The derivative of \( \text{ln}(x) \) is \( \frac{1}{x} \).
• **Behavior:** The natural logarithm function increases slowly and has a vertical asymptote at \( x = 0 \).

Understanding these properties will help you see why \( f(x) = x \, \text{ln}(x) \) behaves the way it does. The function rises quickly initially but then slows down because of the logarithm component.

Knowing the derivative \( \text{ln}(x) \) also helps in applying the product rule, as shown in the derivative section.

Using \( \text{ln}(x) \) in functions is common in many mathematical and real-world applications, such as calculating compound interest and analyzing exponential growth.