Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \log _{10}\left(x^{3}+9\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{3x^2}{(x^3 + 9) \ln(10)} \).

Step 1: Understanding the Derivative

The task requires finding the derivative of the function \( \log_{10} (x^3 + 9) \) with respect to \( x \). To approach this, recall that the derivative of \( \log_b(u) \) is \( \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \).

Step 2: Differentiate the Inner Function

Identify the inner function \( u = x^3 + 9 \). We first differentiate \( u \) with respect to \( x \), giving \( \frac{du}{dx} = 3x^2 \).

Step 3: Apply the Derivative Formula for Logarithms

Using the formula \( \frac{d}{dx} \log_b(u) = \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \), substitute \( u = x^3 + 9 \), \( \frac{du}{dx} = 3x^2 \), and \( b = 10 \).

Step 4: Substitute and Simplify

By substituting, we get:\[D_x \log_{10}(x^3 + 9) = \frac{1}{(x^3 + 9) \ln(10)} \cdot 3x^2\]Simplifying this gives us the final derivative:\[\frac{3x^2}{(x^3 + 9) \ln(10)}\]

Step 5: Conclusion

The derivative of \( \log_{10}(x^3 + 9) \) with respect to \( x \) is \( \frac{3x^2}{(x^3 + 9) \ln(10)} \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique that can be incredibly useful when finding the derivative of functions that are products or quotients of several components, especially those involving complicated expressions or exponential terms. For instance, when you encounter a logarithmic function such as \( \log_b(u) \), where \( u \) is a function of \( x \), the derivative involves using both the properties of logarithms and differentiation rules.

This method is particularly helpful because it allows you to take advantage of the logarithm's ability to simplify multiplication, division, and exponentiation into addition, subtraction, and multiplication. This means that by taking the logarithm of the entire function first, you can transform the problem into simpler ones, making the differentiation process far more straightforward.

• The key rule to remember is: \( \frac{d}{dx} \log_b(u) = \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \).

By breaking down the function into its components, you can methodically apply this derivative rule to each piece.

Chain Rule

The chain rule is one of the most important tools in differentiation. It allows you to find the derivative of complex functions by differentiating their nested functions separately. This is especially useful in logarithmic differentiation, where the argument of the logarithm is itself a complex function.

Think of the chain rule as a way to "unwrap" these layers. For a function \( f(g(x)) \), the chain rule states that the derivative is \( f'(g(x)) \cdot g'(x) \). In simpler terms, you differentiate the outer function and then multiply it by the derivative of the inner function.

• This rule becomes crucial when applying logarithmic differentiation, where you often have a logarithmic function with a polynomial or more complex expression inside.
• Applying the chain rule in cases like \( \log_{10}(x^3 + 9) \) involves first differentiating the outer logarithmic function, then the inner polynomial \( x^3 + 9 \).

Mastering the chain rule allows for efficient and correct computation of derivatives in compound functions.

Derivative of Logarithmic Functions

Derivatives of logarithmic functions such as \( \log_b(x) \) are derived from the natural logarithm function, making them essential in calculus.

To differentiate \( \log_b(u) \), where \( b \) is the base of the logarithm, remember:

• The derivative of \( \log_b(x) \) is \( \frac{1}{x \ln(b)} \).
• When the argument is a function \( u = f(x) \), use the formula \( \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \).

This formula combines the basic derivative of the logarithm with the chain rule, allowing you to account for the function within the logarithm.

In problems like differentiating \( \log_{10}(x^3 + 9) \), you apply this combined rule to efficiently compute the derivative, emphasizing the importance of understanding the relationship between the inside function's derivative and the outer logarithmic function's derivative. This principle is widely applicable in many calculus problems involving logarithms, enhancing problem-solving capabilities.