Problem 73 - Applied Calculus

Question

Find the derivative of each function. $\ln \left(x^{2}+5 x\right)

Step-by-Step Solution

Answer

The derivative is \( \frac{2x + 5}{x^2 + 5x} \).

Step 1: Identify the function

The given function is the natural logarithm of a composite function: \( f(x) = \ln(x^2 + 5x) \). We need to find the derivative of this function using the chain rule.

Step 2: Apply the chain rule

To differentiate \( \ln(u) \) where \( u = x^2 + 5x \), apply the chain rule. The chain rule states that the derivative of \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). For \( \ln(u) \), the derivative is \( \frac{1}{u} \), so we find \( \frac{1}{x^2 + 5x} \cdot (x^2 + 5x)' \).

Step 3: Differentiate the inner function

Differentiate the inner function \( u = x^2 + 5x \). The derivative of \( x^2 + 5x \) is \( 2x + 5 \).

Step 4: Combine the results

Using the chain rule, multiply the derivative of the logarithmic part by the derivative of the inside function to get the overall derivative: \( \frac{1}{x^2 + 5x} \cdot (2x + 5) \).

Step 5: Simplify the expression

The final derivative simplifies to \( \frac{2x + 5}{x^2 + 5x} \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. A composite function is essentially a function nested within another function, and the chain rule allows us to break it down step-by-step.
To put it simply, if you have a function like \( f(g(x)) \), where \( f \) and \( g \) are both functions of \( x \), the chain rule helps you find the derivative by following these steps:

• Dynamically identify the outer function (\( f \)) and the inner function (\( g \)).
• Differentiate the outer function \( f \) while treating the inner function \( g(x) \) as a single variable. This gives you \( f'(g(x)) \).
• Then, differentiate the inner function \( g(x) \), obtaining \( g'(x) \).
• Finally, multiply these results together: \( f'(g(x)) \cdot g'(x) \).

This is how you "chain" the derivatives together, hence the term "chain rule". This rule is particularly useful when dealing with functions under a power or log, such as \( \ln(x^2 + 5x) \).

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base of Euler's number \( e \), which is approximately 2.71828. It is widely used in calculus because of its nice derivative properties and its presence in exponential growth models.
Here's what makes \( \ln(x) \) and its derivative noteworthy:

• The function \( \ln(x) \) increases slowly; it rises much slower than polynomial functions or exponential functions.
• The derivative of \( \ln(x) \), which is \( \frac{1}{x} \), shows that the slope of the ln function becomes smaller as \( x \) increases, highlighting its logarithmic nature.
• When taking the derivative of \( \ln(u) \), where \( u \) is a function of \( x \), you need to use the chain rule (as shown in the exercise's solution), making \( \ln \) an interesting application for chain differentiation.

In our example, \( \ln(x^2 + 5x) \), the \( x^2 + 5x \) part is handled as the inner function \( u \), and its differentiation is crucial to applying the chain rule correctly.

Differentiation Rules

Differentiation rules are the basic formulas and techniques used in calculus to find derivatives of functions. These rules simplify the process of finding the rate at which a function changes at any point. Here are some essential concepts:

• The power rule for differentiation states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• The sum rule enables us to differentiate two added functions separately, such that \( (f(x) + g(x))' = f'(x) + g'(x) \).
• The derivative of a constant is zero, meaning that constants in a function do not affect the rate of change.

These fundamental rules are bolstered by the chain rule, especially for composite functions.
In our exercise, we encountered both the power rule (for \( x^2 \) and \( 5x \)) and the application of the chain rule (within the \( \ln(x) \) function). By applying these rules, you can break down complex differentiation into manageable parts, ultimately finding the derivative of even the most intricate functions.