91Ó°ÊÓ

When differentiating functions, you'll often encounter constants multiplied by a function. The **constant multiple rule** simplifies our task of differentiation. This rule states that if you have a function of the form \( y = c \times f(x) \), where \( c \) is a constant and \( f(x) \) is a function of \( x \), then the derivative is simply the constant multiplied by the derivative of \( f(x) \).

Let's break this down with an example. Suppose we have:

• \( y = 3 \times f(x) \)

According to the constant multiple rule, its derivative is:

• \( y' = 3 \times f'(x) \)

In our exercise, the function given is \( y = 3 \frac{\text{ln} x}{x} \). Here, \( 3 \) is our constant, and \( \frac{\text{ln} x}{x} \) is our function \( f(x) \). By the constant multiple rule, to find the derivative, we first differentiate \( \frac{\text{ln} x}{x} \) and then multiply the result by 3.

When we deal with fractions where the numerator and the denominator are both functions of \( x \), we use the **quotient rule**. The quotient rule states that if you have a function of form \( \frac{u(x)}{v(x)} \), its derivative is given by:
\[ \frac{d}{dx}\frac{u(x)}{v(x)} = \frac{u'v - uv'}{v^2} \] This ensures that we accurately account for the changes in both the numerator \( u(x) \) and the denominator \( v(x) \).
Here’s how we apply it in our exercise: we are given the function:

• \( \frac{\text{ln} x}{x} \)

Let's identify our numerator and denominator:

• \( u = \text{ln} x \), so \( u' = \frac{1}{x} \)
• \( v = x \), so \( v' = 1 \)

Substituting these into our quotient rule formula, we get:
\[ \frac{u'v - uv'}{v^2} = \frac{\frac{1}{x} \times x - \text{ln} x \times 1}{x^2} = \frac{1 - \text{ln} x}{x^2} \] By correctly applying the quotient rule, we find the derivative is \( \frac{1 - \text{ln} x}{x^2} \).

Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function changes at any given point. The derivative of a function gives us the slope of the tangent line to the function at any point.

In our exercise, we are asked to differentiate the function \( y = 3 \frac{\text{ln} x}{x} \). To differentiate:

• We first identified the constant and applied the constant multiple rule to simplify our task.
• Next, we used the quotient rule to differentiate the fraction \( \frac{\text{ln} x}{x} \).
• Finally, we multiplied our result by the constant 3.

Breaking it down step-by-step: 1. **Identify the function parts**: Determine the constant and the function to be differentiated. 2. **Apply Constant Multiple Rule**: Take out the constant and focus on differentiating the function. 3. **Quotient Rule**: Use the rule for fractions with functions in both numerator and denominator.

This gave us our final derivative:

• \( y' = \frac{3(1 - \text{ln} x)}{x^2} \)

Differentiation allows us to understand how a function behaves at any point, making it a crucial tool in calculus and various real-world applications.