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Chapter 10: Problem 64

In Exercises, find the second derivative of the function. $$ f(x)=\log _{10} x $$

Short Answer

Expert verified
The second derivative of the function \( f(x) = \log_{10}x \) is \( f''(x) = -\frac{1}{x^2 \ln 10} \).

Step by step solution

01

Apply the first derivative

First, take the derivative of the given function \( f(x) = \log_{10}x \). Now, remember that the derivative of \( \log_{10}x \) is \( \frac{1}{x \ln 10} \). So, \( f'(x) = \frac{1}{x \ln 10} \).
02

Apply the second derivative

Next, find the second derivative by differentiating the function obtained in Step 1. Remember, the derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \). So, apply this rule to find the second derivative, \( f''(x) = -\frac{1}{x^2 \ln 10} \).
03

Simplify the result

Finally, simplify the result obtained in Step 2. The final result will be \( f''(x) = -\frac{1}{x^2 \ln 10} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Differentiation
Logarithmic differentiation is a powerful calculus technique used when differentiating complex functions, especially when the function has variables in both the base and the exponent. It鈥檚 particularly useful for functions where the variable is an exponent, as logarithmic properties can then be used to simplify the differentiation process. When you take the logarithm of a function and then differentiate it, properties of logarithms allow you to bring down exponents which can make finding the derivative much simpler than directly differentiating the original function. This method is often paired with the chain rule to handle composite functions efficiently.
Derivative of Logarithmic Functions
When dealing with the derivative of logarithmic functions, it's essential to keep in mind that the derivative of \( \text{{ln}}(x) \), which is the natural logarithm, is \( \frac{1}{{x}} \). For other logarithmic bases, such as \( \text{{log}}_{10}(x) \), we use the change-of-base formula to convert it into a natural logarithm, \( \frac{1}{{x \text{{ln}}(10)}} \). This adaptation accommodates the different base and simplifies differentiation. Understanding these principles is key to solving problems involving the derivatives of logarithms effectively.
Calculus
Calculus is a branch of mathematics that deals with rates of change (differential calculus) and accumulation of quantities (integral calculus). It incorporates many functions, including logarithmic and exponential functions, and provides a framework for solving complex problems in both pure and applied mathematics. Differentiation, a fundamental operation in calculus, measures how a function changes as its input changes. Learning calculus enables you to understand concepts like derivatives, including the first and second derivatives, which reveal the rate of change and the rate at which the rate of change is changing, respectively.
Chain Rule
The chain rule is a crucial differentiation tool in calculus used when dealing with composite functions, where one function is nested within another. It states that to differentiate a composite function, you take the derivative of the outer function and multiply it by the derivative of the inner function. Expressed as an equation, if \( f(x) = h(g(x)) \), then \( f'(x) = h'(g(x)) \times g'(x) \). It simplifies the process of finding derivatives for complicated functions and when paired with other differentiation rules, like logarithmic differentiation, it provides a robust method for tackling a variety of differentiation challenges.

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Most popular questions from this chapter

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{5} 12 $$

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