Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ f^{\prime}(81) \text { if } f(x)=\ln \sqrt[3]{x} $$

Step-by-Step Solution

Answer

The derivative \( f'(81) \) is \( \frac{1}{243} \).

Step 1: Simplify the Function

First, let's rewrite the function. We have \( f(x) = \ln \sqrt[3]{x} \). This can be rewritten using exponent rules as \( f(x) = \ln(x^{1/3}) \).

Step 2: Use the Logarithm Power Rule

Apply the logarithmic identity \( \ln(a^b) = b \ln(a) \) to simplify the function further. Thus, \( f(x) = \ln(x^{1/3}) \) becomes \( f(x) = \frac{1}{3} \ln(x) \).

Step 3: Differentiate the Function

Differentiate \( f(x) = \frac{1}{3} \ln(x) \) with respect to \( x \). The derivative of \( \ln(x) \) is \( \frac{1}{x} \), so by using the constant multiple rule, \( f'(x) = \frac{1}{3} \cdot \frac{1}{x} = \frac{1}{3x} \).

Step 4: Evaluate the Derivative at x = 81

Now plug \( x = 81 \) into the derivative \( f'(x) = \frac{1}{3x} \). We have:\[ f'(81) = \frac{1}{3 \times 81} = \frac{1}{243}. \]

Key Concepts

Logarithm Rules

Understanding logarithm rules is essential to solve problems involving logarithmic functions. One key rule is the power rule for logarithms: \( \ln(a^b) = b \ln(a) \). This rule simplifies expressions where the input of a logarithm is raised to a power. For example, if you have \( \ln(x^{1/3}) \), applying this rule gives you \( \frac{1}{3} \ln(x) \). This conversion makes it easier to handle the derivative later on.
Additionally, remember that \( \ln(\sqrt[n]{x}) \) can always be rewritten as \( \ln(x^{1/n}) \), which is especially useful when dealing with square or cube roots. Logarithm rules help simplify functions before applying differentiation techniques, making the overall process much smoother. Always look for such opportunities to simplify logarithmic expressions before differentiation.

Differentiation Techniques

Differentiation is a cornerstone concept in calculus, allowing us to find the rate of change of a function. For logarithmic functions, specifically, knowing the derivative of \( \ln(x) \) is crucial. The basic rule is that the derivative of \( \ln(x) \) is \( \frac{1}{x} \). This helps in calculating derivatives efficiently.
Another common technique is the constant multiple rule, which states that if a function is multiplied by a constant, the derivative is the constant multiplied by the derivative of the function. For instance, if we have \( f(x) = \frac{1}{3} \ln(x) \), the derivative \( f'(x) \) will be \( \frac{1}{3} \) times the derivative of \( \ln(x) \), resulting in \( \frac{1}{3x} \).

• Power rule for differentiation applies to functions involving \( x^n \).
• Chain rule: Useful when dealing with composite functions, though not needed here.

By utilizing these techniques, you can efficiently handle differentiation of various functions.

Evaluation of Derivatives

Evaluating a derivative means finding the specific value of the derivative function at a certain point. In our example, after finding the derivative of the function \( f(x) = \frac{1}{3} \ln(x) \) to be \( f'(x) = \frac{1}{3x} \), we need to evaluate it at \( x = 81 \).
To do this, substitute 81 into the derivative function: \( f'(81) = \frac{1}{3 \times 81} = \frac{1}{243} \).
This step provides the slope of the tangent to the curve of the original function \( f(x) \) at \( x = 81 \).

• Plug the specific \( x \)-value into the derivative function.
• Simplify if necessary to get the exact slope value.

Remember, evaluating derivatives gives insight into the behavior of the function at particular points, like finding the rate at which you're moving along the curve at that instant.