Problem 75 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{4} x+\log _{4} x^{2}$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = \frac{3}{x \ln 4}\).

Step 1: Rewrite the Function Using Logarithm Properties

Apply the properties of logarithms to rewrite the function. Recall that \(\log_a b^c = c \log_a b\). Therefore, the second term \(\log_4 x^2\) can be rewritten as \(2 \log_4 x\). The function becomes:\[y = \log_4 x + 2 \log_4 x = 3 \log_4 x.\]

Step 2: Change of Base Formula

Utilize the change of base formula for logarithms: \(\log_a b = \frac{\ln b}{\ln a}\). Applying this formula to \(3 \log_4 x\) gives:\[y = 3 \frac{\ln x}{\ln 4}.\]

Step 3: Differentiate Using the Chain Rule

Differentiate \(y = 3 \frac{\ln x}{\ln 4}\) with respect to \(x\). The derivative of \(\ln x\) is \(\frac{1}{x}\). Therefore, \[\frac{dy}{dx} = 3 \times \frac{1}{\ln 4} \times \frac{1}{x} = \frac{3}{x \ln 4}.\]

Step 4: Simplify the Derivative

The derivative \(\frac{dy}{dx}\) can be simplified if necessary, but it is already in its simplest form. Thus, the final answer is:\[\frac{dy}{dx} = \frac{3}{x \ln 4}.\]

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique in calculus used to differentiate functions that are complex, such as products or quotients of functions.
This method can simplify the differentiation process by employing properties of logarithms.

When you see a logarithmic expression like in the given problem with base 4, you can apply the property:

• \(\log_a(b^c) = c\log_a(b)\)

This means that the exponent rules for logarithms make it easier to manage terms within the function before differentiation.
For instance, given \( y = \log_4 x + \log_4 x^2 \), this can transform into \( y = \log_4 x + 2\log_4 x \), which then consolidates to \( y = 3 \log_4 x \).

Logarithmic differentiation can turn intricate multiplications into simple additions. This is useful when integrating or derivating complicated expressions.

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions.
Simply put, when you have a function within another function, the outer function's derivative is taken with respect to the inner function, and then multiplied by the inner function's derivative.

For the function \( y = 3 \frac{\ln x}{\ln 4} \), recognize that \( \ln x \) is inside the larger constant multiplier \(\frac{3}{\ln 4}\).

• The derivative of \( \ln x \) is \( \frac{1}{x} \).

Thus, you apply the chain rule by keeping the constant outside and differentiating \( \ln x \) directly.
Ultimately, this results in the derivative: \( \frac{dy}{dx} = 3 \times \frac{1}{\ln 4} \times \frac{1}{x} \). This technique is essential in deriving complex nested functions.

Change of Base Formula

The change of base formula is a useful trick in handling logarithms, particularly when dealing with different bases.
This formula states that \( \log_a b = \frac{\ln b}{\ln a} \).

In this exercise, the function initially expressed in base 4, \( y = 3 \log_4 x \), can be hard to differentiate directly.
By converting it to base \( e \) using the change of base formula, it becomes \( y = 3 \frac{\ln x}{\ln 4} \), where the natural logarithm is easier to differentiate.

• Natural logarithms (ln) are conventionally easier to work with in calculus.

This conversion simplifies differentiation significantly, as the derivative of \( \ln x \) is straightforwardly \( \frac{1}{x} \).
Thus, the change of base formula acts as a bridge for transforming complex logarithmic derivatives into simpler forms.