Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \log _{3} e^{x} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{\ln(3)} \).

Step 1: Apply the Quotient Rule for Derivative of a Logarithm

The derivative of a logarithm in the form \( \log_b(f(x)) \) is given by \( \frac{1}{f(x) \ln(b)} \cdot f'(x) \). Therefore, for \( D_x \log_3 e^x \), we can write it as: \[ \frac{1}{e^x \ln(3)} \cdot \frac{d}{dx}(e^x). \]

Step 2: Differentiate the Inner Function

The function inside the logarithm, \( f(x) = e^x \), has a derivative \( f'(x) = e^x \). Substitute this derivative back into the expression from Step 1.

Step 3: Substitute and Simplify

Substitute \( f'(x) = e^x \) into the equation from Step 1: \[ \frac{1}{e^x \ln(3)} \cdot e^x = \frac{e^x}{e^x \ln(3)}. \] Simplify the expression: \[ \frac{e^x}{e^x \ln(3)} = \frac{1}{\ln(3)}. \]

Step 4: Final Answer

The derivative \( D_x \log_3 e^x \) simplifies to \( \frac{1}{\ln(3)} \). This is the final answer.

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique often used to differentiate functions where the variable is both in base and in exponent. It can turn complex multiplicative and exponential relationships into simpler additive ones. This becomes especially useful for complex functions such as the product of multiple terms or when dealing with variables in the exponent.
To use logarithmic differentiation, you:

• Take the natural logarithm of both sides of the equation.
• Differentiate implicitly with respect to x.
• Solve for the derivative of the original function.

Applying logarithmic differentiation to problems involving logarithmic and exponential functions capitalizes on the properties of logarithms to simplify the differentiation process. This method was used in the provided exercise to handle the logarithmic function by applying the rule \( \frac{d}{dx} \log_b f(x) = \frac{1}{f(x) \ln(b)} \cdot f'(x) \). This rule helps you find the derivative of a logarithmic function with a base other than e.

Calculus

Calculus is the branch of mathematics that studies how things change. It's divided into two main parts: differential calculus and integral calculus. The derivative, which plays a primary role in differential calculus, represents how a function changes as its input changes.
The process of finding a derivative is called differentiation. Derivatives have various practical applications, from physics to economics, wherever rates of change are important. In the given exercise, calculus is used to find the derivative of the function \( \log_3 e^x \).
Using derivatives:

• Helps calculate the rate of change in real-world situations.
• Assists in finding slopes of tangent lines to curves.
• Is essential in understanding motion and change.

In this specific case, the derivative rules for logarithms and exponentials are applied step by step. These rules are vital since they simplify the derivative calculation for complex functions, as illustrated in the earlier steps of the solution.

Exponential Functions

Exponential functions are a type of mathematical function where the variable is in the exponent position. This makes them grow rapidly and frequently appear in real-world phenomena, such as population growth and radioactive decay. The most common exponential function is the natural exponential function \( e^x \), where e is the base of the natural logarithm, approximately equal to 2.71828.
Properties of exponential functions include:

• Their derivative is proportional to the value of the function itself.
• They have a constant rate of growth or decay.
• They're used to model phenomena involving continuous growth.

In calculus, differentiating exponential functions is straightforward since the derivative of \( e^x \) is itself \( e^x \). The exercise demonstrates this principle, as differentiation of the function inside the logarithm is required. Understanding exponential functions and how to differentiate them is crucial, as they often form part of more complex equations, whether in mathematical studies or in analyzing real-world scenarios.