Problem 6 - Calculus Single Variable

Question

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(x \ln (x)\)

Step-by-Step Solution

Answer

The first derivative is \(1 + \ln(x)\), the second is \(\frac{1}{x}\), and the third derivative is \(-\frac{1}{x^2}\).

Step 1: Understanding the Function

The function given is \( f(x) = x \ln(x) \). It is a product of two functions: \( x \) and \( \ln(x) \). We need to use the product rule to differentiate this expression.

Step 2: Applying the Product Rule for the First Derivative

The product rule states that if you have two functions \( u \) and \( v \), then the derivative is \( uv' + vu' \). For our function, let \( u = x \) and \( v = \ln(x) \). The derivative of \( u \) is \( 1 \), and the derivative of \( v \) is \( \frac{1}{x} \). Therefore, the first derivative \( f'(x) = x \cdot \frac{1}{x} + \ln(x) \cdot 1 = 1 + \ln(x) \).

Step 3: Differentiating Again for the Second Derivative

The first derivative is \( f'(x) = 1 + \ln(x) \). Differentiating \( 1 \) gives \( 0 \), and differentiating \( \ln(x) \) gives \( \frac{1}{x} \). Thus, the second derivative is \( f''(x) = 0 + \frac{1}{x} = \frac{1}{x} \).

Step 4: Differentiating Once More for the Third Derivative

From the second derivative \( f''(x) = \frac{1}{x} \), differentiating \( \frac{1}{x} \) gives \( f'''(x) = -\frac{1}{x^2} \), which uses the power rule.

Key Concepts

Product Rule

When we are dealing with the differentiation of a product of two functions, the product rule comes into play. If a function is expressed as a product of two other functions, say \( u \) and \( v \), then according to the product rule, the derivative is determined as follows:

• The derivative of \( u \) is multiplied by \( v \).
• The derivative of \( v \) is multiplied by \( u \).
• Add these two products together.

So, if you have a function \( f(x) = u(x) \cdot v(x) \), its derivative \( f'(x) \) using the product rule is: \[ f'(x) = u'(x)v(x) + u(x)v'(x) \]This rule is exceptionally useful when handling complex functions made from basic mathematical operations. It allows us to systematically take the derivative of each component and combine them.

First Derivative

The first derivative of a function provides insight into its slope or rate of change. For our given function \( f(x) = x \ln(x) \), applying the product rule involves the following:

• Let \( u = x \) and \( v = \ln(x) \).
• The derivative of \( u \), which is \( x \), is \( 1 \).
• The derivative of \( v \), which is \( \ln(x) \), is \( \frac{1}{x} \).
• Plugging these into the product rule formula gives us: \( x \cdot \frac{1}{x} + \ln(x) \cdot 1 = 1 + \ln(x) \).

So, the first derivative \( f'(x) \) is \( 1 + \ln(x) \). This tells us how the function \( f(x) = x \ln(x) \) behaves locally. For small changes around a point, this derivative indicates the speed of change and the direction whether the function is increasing or decreasing.

Second Derivative

The second derivative of a function is significant as it describes the curvature or concavity of the function. For our function \( f(x) = x \ln(x) \), we have already calculated the first derivative as \( f'(x) = 1 + \ln(x) \). The second derivative is obtained by differentiating this result:

• The derivative of \( 1 \) is \( 0 \) because constants do not change.
• The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
• This results in the second derivative \( f''(x) = 0 + \frac{1}{x} = \frac{1}{x} \).

The second derivative \( f''(x) = \frac{1}{x} \) gives us information on the concave or convex nature of \( f(x) \). If \( x > 0 \), then \( f''(x) > 0 \), indicating the function is concave up.

Third Derivative

Moving further to the third derivative, we can find insights into the "jerk" of the function's graph, which means the rate of change of the curvature. Starting from the second derivative \( f''(x) = \frac{1}{x} \), the third derivative is calculated as follows:

• The derivative of \( \frac{1}{x} \) can be found using the power rule.
• Recall that \( \frac{1}{x} \) can be expressed as \( x^{-1} \).
• Using the power rule, the derivative of \( x^{-1} \) is \( -x^{-2} \), or \( -\frac{1}{x^2} \).

Thus, the third derivative \( f'''(x) = -\frac{1}{x^2} \). This measure can be useful in more advanced analyses of the function, especially when considering motion or higher-dimensional contexts, which help in understanding how curvature itself changes.