Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \ln \left(x^{2}+3 x+\pi\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{2x + 3}{x^2 + 3x + \pi} \).

Step 1: Identify the Function and Derivative Rule

The function we need to differentiate is \( \ln(x^2 + 3x + \pi) \). Since it is a natural logarithm function, we'll apply the derivative rule for logarithms: \( D_x[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = x^2 + 3x + \pi \).

Step 2: Differentiate the Inner Function

We first need to find the derivative of the inner function \( u = x^2 + 3x + \pi \). Differentiating term by term gives: \( \frac{du}{dx} = 2x + 3 \).

Step 3: Apply the Chain Rule for Logarithmic Differentiation

Using the derivative rule for logarithms, substitute the inner function and its derivative into the formula: \( D_x[\ln(x^2 + 3x + \pi)] = \frac{1}{x^2 + 3x + \pi} \cdot (2x + 3) \).

Step 4: Simplify the Expression

Combine the terms to express the derivative: \( \frac{2x + 3}{x^2 + 3x + \pi} \). This is the derivative of the given function.

Key Concepts

Chain Rule

The chain rule is a vital tool in calculus for finding the derivative of composite functions. A composite function is when one function is nested inside another, like peeling layers from an onion. When differentiating such functions, we apply the chain rule to break it down into simpler components. In simpler terms:

• Differentiate the outer function with respect to the inner function.
• Multiply this by the derivative of the inner function.

For example, in the exercise where we differentiate \( \ln(x^2 + 3x + \pi) \), \( \ln(u) \) is the outer function, and \( u = x^2 + 3x + \pi \) is the inner function. To apply the chain rule here, we1. Differentiate the natural logarithm, giving us \( \frac{1}{u} \).2. Multiply by the derivative of the inner function, \( \frac{du}{dx} = 2x + 3 \).3. Combine these results to obtain the derivative: \( \frac{1}{x^2 + 3x + \pi} \times (2x + 3) \).This efficiently captures the transition from composite to simple differentiation.

Natural Logarithm

Natural logarithm functions involve the constant \( e \), which is approximately 2.71828. The natural logarithm, noted as \( \ln(x) \), has unique properties that make differentiation manageable. The derivative of \( \ln(x) \) is \( \frac{1}{x} \), a crucial fact which simplifies many calculations in calculus. In the context of a function like \( \ln(x^2 + 3x + \pi) \), the natural logarithm allows us to use the chain rule effectively by setting \( u = x^2 + 3x + \pi \) and calling upon the derivative rule \( D_x[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \). This rule asserts that the logarithm's peculiarity lies in its derivative transforming a complex function into its reciprocal times the inner derivative. Thus, understanding the behavior of \( \ln(x) \) is essential, as it facilitates handling intricate expressions—a powerful technique in calculus.

Differentiation Rules

Differentiation rules, such as those for powers, products, and sums, provide the backbone for calculating derivatives swiftly. These rules guide us in methodically finding derivatives of basic parts that make up complex functions.Some core differentiation rules include:

• The power rule: For \( x^n \), the derivative is \( nx^{n-1} \).
• The sum rule: The derivative of a sum is the sum of the derivatives: \( D_x[f(x) + g(x)] = f'(x) + g'(x) \).
• The constant rule: The derivative of a constant is zero.

In our exercise, these rules help deduce the derivative of the polynomial \( x^2 + 3x + \pi \) as each term is differentiated separately: \( 2x \) for \( x^2 \) and \( 3 \) for \( 3x \), while \( \pi \) being a constant remains zero. Together, these differentiation rules provide the basic toolkit for solving calculus problems, enabling effective usage of more advanced rules like the chain rule.