Problem 52 - Calculus of a Single Variable

Question

In Exercises 41–64, find the derivative of the function. $$ f(x)=\ln \left(\frac{2 x}{x+3}\right) $$

Step-by-Step Solution

Answer

The derivative of the given function is \( \frac{1}{x} - \frac{1}{x + 3} \).

Step 1: Apply Properties of Logarithms

Rewrite the function using properties of logarithms to simplify. The log of a quotient (division) can be rewritten as the difference of the logs. Hence, \( f(x) = \ln 2x - \ln (x + 3) \).

Step 2: Apply the Chain Rule

Next, apply the chain rule. Recall that the derivative of \( ln(u) \) is \( \frac{1}{u} * u' \), where \( u' \) is the derivative of \( u \). Applying this to our terms, we get, \( f'(x) = \frac{1}{2x}*2 - \frac{1}{x + 3}*1 \)

Step 3: Simplify the Result

After simplifying \( f'(x) \), it is found that \( f'(x) = \frac{1}{x} - \frac{1}{x + 3} \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial for working with logarithmic functions, particularly when taking derivatives. Logarithmic properties can transform complex expressions into forms that are easier to differentiate. One fundamental property is the quotient rule: the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.

In the exercise at hand, the function given is \( f(x) = \ln\left(\frac{2x}{x+3}\right) \). By applying the property mentioned above, we split the single logarithm of a division into the subtraction of two separate logs: \( f(x) = \ln(2x) - \ln(x + 3) \). This initial step is key as it simplifies the function, making the upcoming steps of differentiation more straightforward.

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. In essence, if a function \( y \) can be written as \( u(g(x)) \) where both \( u \) and \( g \) are differentiable functions, then the derivative of \( y \) with respect to \( x \) is \( u'(g(x)) \cdot g'(x)\).

In our example, after applying logarithmic properties, we find that the derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u' \) where \( u' \) is the derivative of \( u \). For the term \( \ln(2x) \), using the chain rule gives us \( \frac{1}{2x} \cdot 2 \) because the derivative of \( 2x \) with respect to \( x \) is simply \( 2 \). Similarly, the derivative of \( \ln(x+3) \), is \( \frac{1}{x+3} \cdot 1 \) since the derivative of \( x+3 \) is \( 1 \). The chain rule allows us to differentiate complex functions like these by breaking them down into simpler components.

Simplifying Derivatives

Simplifying the result of a derivative can often reveal a more elegant and useful form. After differentiation, the individual terms can often be combined or reduced to create a simpler expression. In our exercise, after applying the chain rule, we obtained the derivative \( f'(x) = \frac{1}{2x} \cdot 2 - \frac{1}{x + 3} \cdot 1 \). Each term simplifies by multiplying through any constants and removing unnecessary ones.

The term \( \frac{1}{2x} \cdot 2 \) simplifies to \( \frac{1}{x} \), while \( \frac{1}{x + 3} \cdot 1 \) remains \( \frac{1}{x + 3} \). The final, simplified derivative then becomes \( f'(x) = \frac{1}{x} - \frac{1}{x + 3} \). Such a step is important for a clear understanding and to effectively work with the results, especially in more complex applications such as solving equations or evaluating limits. Simplification ensures the easiest form of the derivative for further use.