Problem 21 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\ln ^{2}(x)\)

Step-by-Step Solution

Answer

The derivative of \(\ln^2(x)\) is \(\frac{2\ln(x)}{x}\).

Step 1: Recognize the Outer Function

The expression is given as \(\ln^2(x)\). This is equivalent to \((\ln(x))^2\), where \(\ln(x)\) is inside a square function. The outer function is \(u^2\), where \(u = \ln(x)\).

Step 2: Apply the Chain Rule

To differentiate \((\ln(x))^2\) using the chain rule, we need to apply the derivative of the outer function followed by the derivative of the inner function. The chain rule states if \(f(u) = u^n\) and \(u=g(x)\), then \(\frac{d}{dx}f(g(x)) = n \cdot g(x)^{n-1} \cdot g'(x)\).

Step 3: Compute the Derivative of the Outer Function

The derivative of \(u^2\) with respect to \(u\) is \(2u\). For our expression, this means the derivative of \((u)^2\) with respect to \(u\) is \(2\ln(x)\).

Step 4: Compute the Derivative of the Inner Function

The inner function is \(\ln(x)\). The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\).

Step 5: Combine Derivatives Using the Chain Rule

Using the chain rule, combine the results: \(2\ln(x) \cdot \frac{1}{x}\). This simplifies to \(\frac{2\ln(x)}{x}\).

Step 6: Conclusion

The derivative of \(\ln^2(x)\) with respect to \(x\) is \(\frac{2\ln(x)}{x}\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that helps us differentiate composite functions. Think of it as a way to unstitch layers of a function step by step. When you have an outer function and an inner function combined, the chain rule allows you to find the derivative effortlessly. In our exercise, the expression \( \ln^2(x) \) can be broken down into an outer function \( (\ln(x))^2 \). Here, the outer function is raised to the power of 2, and the inner function is the logarithmic function \( \ln(x) \).

To apply the chain rule, you need to follow these steps:

• Identify the outer and inner functions. Here, the outer is \( u^2 \) where \( u = \ln(x) \).
• Differently the outer function with respect to \( u \) and then multiply it by the derivative of the inner function with respect to \( x \).
• Combine these derivatives as shown in the chain rule formula.

By using this rule, you can differentiate not only logarithmic functions but any composite functions you may encounter.

Logarithmic Functions

Logarithmic functions, particularly the natural logarithm \( \ln(x) \), are deeply intertwined with calculus because they appear in many natural phenomena. Understanding how to differentiate logarithmic functions is crucial for mastering calculus problems.

In our exercise, the function at the core is \( \ln(x) \). The natural logarithm is the inverse of the exponential function and possesses unique differentiation properties:

• The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \).
• Logarithmic functions simplify multiplicative processes into additive processes, which is very useful when differentiating more complex functions.

When you encounter expressions involving \( \ln(x) \), always remember that they can be simplified using properties of logarithms before applying differentiation, making your calculations more manageable.

Differentiation Steps

The steps for differentiation help maintain clarity and precision in calculus problems. Let’s recapture the steps used to differentiate \( \ln^2(x) \) to reinforce the differentiation process.

Here's the process broken down:

• Step 1: Recognize the Outer Function - Write \( \ln^2(x) \) as \( (\ln(x))^2 \) and recognize it involves both an outer \((u^2)\) and an inner function \((\ln(x))\).
• Step 2: Apply the Chain Rule - Differentiate the outer function \( u^2 \) with respect to \( u \), resulting in \( 2u \), and multiply by the derivative of \( \ln(x) \), which is \( \frac{1}{x} \).
• Step 3: Combine Derivatives - Integrate these into one to find \( 2\ln(x) \cdot \frac{1}{x} \), simplifying to \( \frac{2\ln(x)}{x} \).
• Conclusion - You've determined that the derivative of \( \ln^2(x) \) with respect to \( x \) is \( \frac{2\ln(x)}{x} \).

By following these clear steps, you can tackle similar differentiation problems with confidence and precision.