Problem 12 - Single Variable Calculus

Question

Differentiate the function. $$ h(t)=\frac{\ln t}{\ln 2 t} $$

Step-by-Step Solution

Answer

The derivative of the function \(h(t) = \frac{\ln t}{\ln(2t)}\) is: \(h'(t) = \frac{\ln(2t) - \ln t}{t(\ln(2t))^2}\)

Step 1: Identify the function in the quotient rule format

We have the given function \(h(t) = \frac{\ln t}{\ln(2t)}\). We can view this function as a quotient: - Numerator function: \(g(t) = \ln t\) - Denominator function: \(h(t) = \ln(2t)\)

Step 2: Find the derivatives of the numerator and the denominator functions

We need to find the derivatives of the natural logarithm functions in the numerator and the denominator. - Derivative of the numerator function (\(g(t) = \ln t\)): Using the chain rule of differentiation, we have \(g'(t) = \frac{1}{t}\) - Derivative of the denominator function (\(h(t) = \ln(2t)\)): Again, using the chain rule, we have \(h'(t) = \frac{1}{2t} \cdot 2 = \frac{1}{t}\)

Step 3: Apply the quotient rule

Now, we apply the quotient rule of differentiation to find the derivative of the given function: \[ h'(t) = \frac{g'(t)h(t)-g(t)h'(t)}{h(t)^2} \] Putting the derivatives of the numerator and denominator functions and their original functions into the equation: \[ h'(t) = \frac{\frac{1}{t} \cdot \ln(2t) - \ln t \cdot \frac{1}{t}}{(\ln(2t))^2} \]

Step 4: Simplify the expression

We can simplify the expression by factoring out a common \(\frac{1}{t}\) term from the numerator: \[ h'(t) = \frac{\frac{1}{t} (\ln(2t) - \ln t)}{(\ln(2t))^2} \] So, the derivative of the function is: \[ h'(t) = \frac{\ln(2t) - \ln t}{t(\ln(2t))^2} \]

Key Concepts

Quotient Rule

The quotient rule is a fundamental technique in calculus used for differentiating functions that are expressed as the division of two functions. If you have a function that takes the form \( \frac{u(t)}{v(t)} \), the quotient rule helps in finding the derivative of this function. The formula is:

• \( \frac{d}{dt} \left( \frac{u(t)}{v(t)} \right) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \)

In the original exercise, the function \( h(t) = \frac{\ln t}{\ln(2t)} \) fits perfectly into the quotient rule format. Here:

• Numerator function \( g(t) = \ln t \)
• Denominator function \( h(t) = \ln(2t) \)

Using the identified derivatives of these parts, the quotient rule allows us to find:

• The derivative of the complete function is calculated by substituting each function and its derivative into the rule, simplifying any expression if possible.

Chain Rule

To differentiate complex functions, such as those involving compositions of two or more functions, the chain rule is a powerful tool. It states that if you have a composite function \( f(g(t)) \), the derivative is:

• \( f'(g(t))g'(t) \)

In simpler terms, you differentiate the outer function with respect to the inner function, and multiply by the derivative of the inner function.
For the exercise we have, the chain rule is pivotal for differentiating the natural logarithmic expressions:

• The numerator \( \ln t \) gives a direct derivative \( \frac{1}{t} \) since it's a simple natural log derivative.
• The denominator \( \ln(2t) \) necessitates the chain rule. Here, differentiating as \( \ln(2t) \) involves understanding that \( 2t \) is an inner function, whose derivative is \( 2 \). Then, applying chain rule computations result in the derivative \( \frac{1}{2t} \cdot 2 = \frac{1}{t} \).

Natural Logarithm Differentiation

Differentiating natural logarithm functions involves specific rules. The natural logarithm function, denoted by \( \ln(t) \), frequently appears in calculus problems. The basic derivative rule for natural logs is:

• If \( f(t) = \ln(t) \), then \( f'(t) = \frac{1}{t} \).

This rule simplifies the differentiation process and is applied directly to the numerator in the given function \( h(t) = \frac{\ln t}{\ln(2t)} \).
When the logarithm involves more than a simple \( t \), as in \( \ln(2t) \), the chain rule is also useful together with the natural log derivative rule. The procedure always derives from the foundation that inside a logarithm, every variable factor affects the differentiation.

• So, for \( \ln(2t) \), treat \( 2t \) as a unit, and not only differentiate \( \ln \), but also apply differentiation to \( 2t \), allowing the composite derivative to emerge.