Problem 86 - Calculus and its applications

Question

Differentiate. $$ g(x)=(\ln x)^{3} $$

Step-by-Step Solution

Answer

The derivative is \( g'(x) = \frac{3(\ln x)^2}{x} \).

Step 1: Identify the Chain Rule

The function \( g(x) = (\ln x)^3 \) involves a composition of functions. To differentiate it, we will use the chain rule, which is suitable for derivatives of composite functions. The outer function is \( u^3 \) where \( u = \ln x \).

Step 2: Differentiate the Outer Function

Differentiate the outer function \( u^3 \) with respect to \( u \). The derivative is \( 3u^2 \). Here, \( u = \ln x \), so the derivative of the outer function in terms of \( u \) is \( 3(\ln x)^2 \).

Step 3: Differentiate the Inner Function

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

Step 4: Apply the Chain Rule

Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives: \[ g'(x) = 3(\ln x)^2 \cdot \frac{1}{x} = \frac{3(\ln x)^2}{x}. \]

Step 5: Simplify Result

The expression \( \frac{3(\ln x)^2}{x} \) is already in simplest form, representing the final differentiated form of \( g(x) \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus. It's used to differentiate composite functions. Composite functions are functions inside other functions, like layers of an onion. The chain rule helps us peel back these layers to find the derivative. This is handy when differentiating functions like

• \((\ln x)^3\)
• \((3x^2+4)^5\)

The chain rule states that to differentiate a composite function, you differentiate the outer function and multiply it by the derivative of the inner function. If you have a function \( f(g(x)) \), you first differentiate \( f \) with respect to \( g(x) \), and then differentiate \( g(x) \) with respect to \( x \). Finally, multiply these two derivatives together. In our example, we have to differentiate the outer function, \( u^3 \), where \( u = \ln x \), making sure to follow through with the derivative of \( u \) afterwards.

Composite Functions

Composite functions are an essential element of calculus that require careful attention when differentiating. These functions are built by composing two or more functions together. Imagine taking a base function and layering another function on top of it. For example, with our exercise \( g(x) = (\ln x)^3 \), the base or inner function is \( \ln x \) and raising it to the power of three adds the outer layer.
To identify composite functions, look for expressions where one function is nested within another. In these cases, utilize the chain rule to find the derivative. By successfully tackling composite functions, you'll enhance your ability to handle various differentiation problems in calculus.

Logarithmic Differentiation

Logarithmic differentiation is a powerful tool when dealing with functions that involve logarithms, especially when these functions are multiplied or raised to powers.
For complicated expressions where direct differentiation isn't straightforward, logarithmic differentiation simplifies the process by utilizing properties of logarithms.

• Start by taking the natural logarithm of both sides of an equation
• Use log properties to simplify the expression
• Differentiate both sides, applying standard rules

In our example \( g(x) = (\ln x)^3 \), we used the properties of logs to express and differentiate the function effectively using chain and standard differentiation methods. This approach not only simplifies complex derivatives but also provides a strategic method to tackle intricate functions.