Problem 8 - Thomas Calculus in SI Units

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln \left(t^{3 / 2}\right)$$

Step-by-Step Solution

Answer

The derivative of \(y\) with respect to \(t\) is \(\frac{3}{2t}\).

Step 1: Recall the Derivative of a Logarithm

The derivative of \( \log(x) \) with respect to \(x\) is \(\frac{1}{x} \). Since we have \(\ln(t^{3/2})\), we are looking for the derivative of a natural logarithm.

Step 2: Simplify the Expression

Use the property of logarithms that \(\ln(a^b) = b \cdot \ln(a)\). Therefore, \(\ln(t^{3/2}) = \frac{3}{2} \cdot \ln(t)\). Thus, the expression for \(y\) simplifies to \(\frac{3}{2} \ln(t) \).

Step 3: Differentiate the Simplified Expression

Differentiate \(y = \frac{3}{2} \ln(t)\) with respect to \(t\). The derivative of \(\ln(t)\) is \(\frac{1}{t}\), so \(\frac{d}{dt}[y] = \frac{3}{2} \cdot \frac{1}{t}\).

Step 4: Write the Final Derivative

The derivative of \(y = \ln(t^{3/2})\) with respect to \(t\) is \(\frac{3}{2t}\).

Key Concepts

Natural Logarithm

The natural logarithm is an essential concept in calculus and mathematical analysis. It is denoted by "ln" and has its base as Euler's number, approximately 2.718. Natural logarithms have many useful properties which simplify mathematical computations. For example, the natural logarithm of a product is the sum of the natural logarithms of its factors. Furthermore, if we take the natural logarithm of a power, such as \( ext{ln}(a^b) \), this can be simplified to \( b \cdot ext{ln}(a) \). This specific property makes it much easier to differentiate functions involving logarithms.

Understanding the basics of natural logarithms can significantly aid in solving calculus problems.

Properties of Logarithms

Logarithms possess properties that help in simplifying complex expressions. These properties usually involve operations like multiplication, division, and exponentiation. Knowing these properties is critical, especially when differentiating logarithmic functions.

Some crucial logarithmic properties include:

• Product Rule: \( ext{ln}(xy) = ext{ln}(x) + ext{ln}(y) \)
• Quotient Rule: \( ext{ln}(\frac{x}{y}) = ext{ln}(x) - ext{ln}(y) \)
• Power Rule: \( ext{ln}(a^b) = b \cdot ext{ln}(a) \)

The Power Rule was particularly useful in the provided exercise, allowing us to transform \( ext{ln}(t^{3/2}) \) into a more manageable expression, \( \frac{3}{2} \cdot ext{ln}(t) \), before performing differentiation. This simplification step is pivotal before proceeding with derivative calculations.

Differentiation Rules

Differentiation, or finding the derivative, is a fundamental tool in calculus that helps determine how a function changes with respect to its variables. When dealing with logarithmic functions, specific rules apply that make differentiation straightforward.

For the natural logarithm, the derivative of \( ext{ln}(x) \) with respect to \( x \) is \( \frac{1}{x} \), which is a crucial rule to remember. This allows us to differentiate more complex expressions involving logarithms.

In our exercise, we used this rule to differentiate \( y = \frac{3}{2} \, \text{ln}(t) \). Since the derivative of \( \text{ln}(t) \) is \( \frac{1}{t} \), the result is \( \frac{3}{2} \cdot \frac{1}{t} \). Therefore, the derivative of the original function \( y = \text{ln}(t^{3/2}) \) with respect to \( t \) is \( \frac{3}{2t} \). Understanding such straightforward rules helps in solving derivatives quickly and accurately.