Problem 48 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(37-58\) , find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) $$ y=\log _{3}\left(x^{2}-3 x\right) $$

Step-by-Step Solution

Answer

The derivative of the function \(y=\log _{3}\left(x^{2}-3 x\right)\) is \(y' = {1 \over ln(3) * x}\)

Step 1: Apply Logarithmic Properties

Take the given function \(y=\log _{3}\left(x^{2}-3 x\right)\) and apply the logarithm rule \(\log (a * b) = \log(a) + \log(b)\), which allows the expression inside the logarithm to be split into two addends. But here it’s a subtraction, so we can write it as: \(y = \log_{3}(x^2) - \log_{3}(3x)\)

Step 2: Differentiate

The next step is to perform the derivative. The derivative of \(\log_{b}(a)\) is \((1/(\ln(b) * a)) * a'\), where a' is the derivative of 'a'. Therefore we differentiate \(y = \log_{3}(x^2) - \log_{3}(3x)\) using this rule. This gives \(y' = {2x \over ln(3) * x^2} - {3 \over ln(3) * 3x}\)

Step 3: Simplify the Result

Misuse of factors leads to a simpler form of the derivative. After cancelling common factors, the derivative simplifies to \(y' = {2 \over ln(3) * x} - {1 \over ln(3) * x}\) Then, combining these into one fraction results in \(y' = {1 \over ln(3) * x}\)

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique that can make the process of finding derivatives easier, especially for complex expressions. It is particularly useful when dealing with functions that involve products, quotients, or powers. When applying logarithmic differentiation, we take the logarithm of both sides of the function equation, often resulting in a simplified expression before differentiating.

Using the logarithm helps by transforming multiplication into addition and division into subtraction. This transformation is governed by the properties of logarithms and can be crucial for simplifying difficult derivatives. Once we have taken the logarithm and manipulated the expression, we can differentiate both sides using standard differentiation techniques.

Logarithmic Properties

Logarithmic properties are key tools in simplifying and solving expressions involving logarithms. The properties most often used are:

• The Product Rule: \(\log(a * b) = \log(a) + \log(b)\)
• The Quotient Rule: \(\log(a / b) = \log(a) - \log(b)\)
• The Power Rule: \(\log(a^b) = b*\log(a)\)

These properties allow us to break down complex logarithmic expressions into more manageable pieces. In our exercise, we used the quotient rule to split the function \(y = \log_{3}(x^2 - 3x)\) into \(\log_{3}(x^2) - \log_{3}(3x)\). This results in separate components that are simpler to differentiate. By applying these rules, the derivative process becomes clearer and often less prone to mistakes.

Simplifying Derivatives

Simplifying derivatives can greatly help in clarifying the result and ensuring accuracy. Often during differentiation, you may end up with complex fractions or expressions. Simplifying these expressions provides a cleaner and more understandable result.

In our solution, after applying the logarithmic derivative rules, we ended up with an expression that looked like this: \(y' = \frac{2x}{\ln(3) * x^2} - \frac{3}{\ln(3) * 3x}\). To simplify, we checked for common factors that could be canceled out. By reducing these fractions, we combined them into a single term: \(y' = \frac{1}{\ln(3) * x}\). This simplified form is easier to work with and interpret. Always look to combine and reduce expressions where possible to make derivatives manageable.