Problem 9 - Thomas Calculus

Question

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \left(t^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \(\frac{2}{t}\).

Step 1: Identify the type of function

The given function is \(y = \ln(t^2)\). This is a logarithmic function, specifically the natural logarithm of \(t^2\).

Step 2: Apply the logarithmic property

Use the property of logarithms: \(\ln(a^b) = b\ln(a)\). Thus, we can rewrite the function as \(y = 2\ln(t)\).

Step 3: Differentiate the function

Differentiate \(y = 2\ln(t)\) with respect to \(t\). The derivative of \(\ln(t)\) with respect to \(t\) is \(\frac{1}{t}\). Thus, the derivative is:\[\frac{dy}{dt} = 2 \cdot \frac{1}{t} = \frac{2}{t}\]

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a useful technique in calculus that is applied when differentiating functions that are products or quotients of simpler functions, or even more complex combinations like power functions where the exponent varies with the base. This method makes use of the properties of logarithms to simplify differentiation.
In essence, by applying logarithmic differentiation, we transform the original expression into a form that is easier to differentiate.

In this exercise, logarithmic differentiation played a critical role in finding the derivative of the function \( y = \ln(t^2) \).
We used the property of logarithms that states \( \ln(a^b) = b\ln(a) \), thereby transforming the function into a simpler form: \( y = 2\ln(t) \).
This transformation makes the differentiation process more straightforward, as it reduces a complex logarithmic function to a linear multiple of a simpler logarithmic function. After this transformation, we can easily apply basic differentiation rules to solve the problem.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm with a special base \( e \). The number \( e \) (approximately equal to 2.71828) is an irrational constant that arises frequently in mathematics, particularly in calculus.

Natural logarithms have some important properties that make them very versatile and useful:

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
• They allow for the simple transformation of exponents into a format that is easy to work with (like turning multiplication into addition).
• They fulfill the logarithmic identities such as \( \ln(a \times b) = \ln(a) + \ln(b) \) and \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).

In this exercise, the natural logarithm simplifies \( \ln(t^2) \) using the property \( \ln(a^b) = b\ln(a) \), which transformed our problem into a simple linear equation \( 2\ln(t) \), making differentiation easier.

Differentiation Rules

When finding the derivative of functions in calculus, we use a set of rules known as differentiation rules. These rules help us determine the slope of the tangent line to the graph of a function at any point.
Different rules apply to different types of functions, and they include:

• Power Rule: If \( y = t^n \), then \( \frac{dy}{dt} = nt^{n-1} \).
• Product Rule: If \( y = u(t) \cdot v(t) \), then \( \frac{dy}{dt} = u'(t)v(t) + u(t)v'(t) \).
• Quotient Rule: If \( y = \frac{u(t)}{v(t)} \), then \( \frac{dy}{dt} = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \).
• Chain Rule: If \( y = f(g(t)) \), then \( \frac{dy}{dt} = f'(g(t)) \cdot g'(t) \).

For the problem \( y = \ln(t^2) \), we applied the derivative of the natural logarithm, which is \( \frac{1}{t} \), directly thanks to the simple form of \( 2\ln(t) \).
This derivation illustrates the use of basic rules effectively, notably the power of logarithmic transformation to ease the differentiation process.