Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

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: Identify the Function

The function given is \( \log_{10}(x^3 + 9) \). This is a logarithm base 10 of the expression \( x^3 + 9 \).

Step 2: Recall the Derivative Rule for Logarithms

The derivative of \( \log_{a}(u) \) is \( \frac{1}{u \ln a} u' \). Here, \( a = 10 \) and \( u = x^3 + 9 \). The natural logarithm of 10 is abbreviated as \( \ln(10) \).

Step 3: Differentiate the Inner Function

Find the derivative of the inner function \( u = x^3 + 9 \). The derivative \( u' \) is \( 3x^2 \).

Step 4: Apply the Derivative Formula

Use the formula from Step 2 to find the derivative: \[ D_{x} \log_{10}(x^3 + 9) = \frac{1}{(x^3 + 9) \ln 10} \cdot 3x^2 \].

Step 5: Simplify the Expression

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

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to find the derivatives of functions that are composed of variables with exponents, or products, quotients, and powers where basic rules aren't convenient. It's particularly handy when you deal with log-based expressions. For instance, consider the function \( \log_{10}(x^3 + 9) \).
By applying the derivative rule for logarithms, we recognize that the derivative involves two parts: the derivative of the logarithm itself and the derivative of its argument, often called the "inner function." This is where logarithmic differentiation really shines. You first differentiate the outer function, then the inner, neatly combining both to derive the result. This otherwise complex process becomes much simpler, thanks to this method.
When you encounter a logarithmic function, remember:

• Apply the logarithm derivative rule: \( \frac{1}{u \ln(a)} \cdot u' \)
• Find the derivative of the inner function \( u \)

Chain Rule

The chain rule is a fundamental principle in calculus used when differentiating compositions of functions. It plays a crucial role when you have a function within another function. In the context of our example, \( \log_{10}(x^3 + 9) \), we need the chain rule because the logarithm function and the polynomial function \( x^3 + 9 \) are layered together.
The chain rule simplifies this by stating that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Essentially, you differentiate the outer function while keeping the inner function intact, and then multiply the result by the derivative of the inner function.
Consider these steps for using the chain rule:

• Differentiating the outside function, leaving the inside as it is
• Multiplying by the derivative of the inside function

This method helps manage complex differentiation tasks efficiently by breaking them down into manageable parts.

Calculus

Calculus is the branch of mathematics that studies continuous change and underpins much of modern mathematics and its applied fields. It consists broadly of two main sections: differential calculus and integral calculus. Our discussion here focuses on differential calculus, as it deals specifically with the concept of derivatives.
When we talk about finding the derivative, as in \( D_x \log_{10}(x^3 + 9) \), we're essentially measuring how a function changes as its input changes. Derivatives provide a powerful means to understand the behavior of functions, such as growth rates or slopes of curves at given points.
To sum up, key elements of calculus include:

• Understanding limits, which are foundational for defining derivatives
• The concept of continuous change, crucial for the idea of differentiation
• Techniques like the chain rule and logarithmic differentiation, essential tools in the process

Embracing these concepts opens up a broader understanding of how we model and predict real-world scenarios mathematically.